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In short, what is the path to enlightenment? Where to begin?
If I were to begin meditating, what practical list of advises would you give me?

What is this thing you call enlightenment? Why do you "want" enlightenment? And after you achieve this thing, then what?

Hi Richard, welcome to the site! What you asked is a short question (i.e. without a lot of detail in the question). Some users think your question might be easier to answer, if you added more information: like how do you define 'enlightenment' (what is your goal)? Are you only asking about meditation (is 'meditation' the only path to enlightenment)? How much have you learned of 'Buddhism' already? I don't want to oblige you to answer such questions ... other people think this is already a practical question that can lead to practical advice.

I want to set my mind free without resorting to any extreme that implies self harm, or that explicitly relies on the collection of material posession in order to stabilize my life and thoughts. If there is any metaphysical way to achieve this, that is honest and, above all, real, I'm willing to indulge in learning about it first. The only other reward, if there is one to expect at all, is to be of help to others.

I honestly don't know how else to put it apart from the fact that you need to ask yourself the right questions and learn along your journey in life. The answers are in every moment. How you see it is left to you.

You can only start from where you are. We cannot tell you where this is, so it's a tricky question. Perhaps you could go to a few sessions at a local Buddhist center to kick--start the process.

The shortest explanation of the path to enlightenment is to not crave or cling egoistically (possessively) to anything, which includes meditation & enlightenment. For example, in meditation, if everything becomes peaceful, calm & blissful, the mind does not think: "I am blissful" or "I love this bliss". Instead, the mind just experiences the bliss, without thinking about or judging it.
Similarly, when reading or hearing about 'breathing meditation', the mind should not try to do breathing meditation because this is craving. Instead, the mind should just quietly & gently free itself from craving (while sitting in meditation) and allow the breathing meditation to occur by itself.
While 'non-clinging' is not exactly the same as 'non-judging', for a beginner, it is similar. Therefore, a short path to enlightenment is described as follows:

You should train yourself thus: 'When seeing, merely see what is seen; when hearing, merely hear what is heard; when
  sensing, merely sense what is sensed; when cognizing (a mental
  object), merely cognise what is cognized.' In this way you should
  train yourself, Bahiya.
Bahiya Sutta

Meditation for enlightenment requires the foundation of non-harming; with the aspiration to not harm anything or anyone. This makes & keeps the mind pure, without obstacle or hindrance. It also generates compassion, which is an important energy for meditation. Here, a clear v ision/view in respect to sexual morals is especially important because the primary obstacle to meditation for enlightenment is often sexual desire. When there is non-harmful view about sex, the instinctual sexual drive can be transformed into compassion.
Also, beginning by doing a short meditation retreat (7 to 10 days) in a meditation centre or monastery can be particularly useful; not so much for the teachings; but for the atmosphere.

You would like to know where to begin this Path to Enlightenment, and you want the answer to be short.  The question is whether you are clever enough to have confidence in the Dhamma that Buddha had explained systematically and in detail. If you cannot develop an unflinching belief in the Buddha (Saddha), you will not go far. If Buddha’s advice to you is to first practice the meditations of ‘Asubha Sanna’ (idea of the impurity of the body), Loathsomeness of food, not taking delight in worlds, Considering all Sankharas as impermanent, and thinking of death before getting into Breathing Meditation, but you think otherwise, then you will not go far.
A person who accepts the Buddha’s Teachings  is in it for the long run. He will not be in a hurry.  His will be a slow, continuous, methodical, orderly practice. Even if it takes five to ten years to get results, he will not falter.  If such a person fails to realize Enlightenment (Nibbana) in this life, he will definitely do so within seven lifetimes the most.
When meditating such a person will only have one goal – that is to liberate from this never ending Samsara – the never-ending cycle of re-becoming.  To have this level of confidence you have to ONLY read the TRUE teachings of the Budddha and accept it. There is an inherent calmness in such a person. He is never in a hurry. This calmness is not a forcefully formed one. All these starts with ‘Sadda’ -  the unflinching belief in the Buddha, the Dhamma and the Arya Sangha – the Arahants.

I really like the non compromising disciplinary approach you speak of. However, even though I have been witness to experiences that strongly suggest to me the notion of the spiritual points to something that is real, I am not convince of the concept of Samsara. This makes me wonder if it is absolutely essential for me to embrace such an idea without evidence; or if not, will my journey towards enlightenment still be a worthwhile one.

Try to develop confidence in the Buddha, and His Teachings @Richard Santiago. As Buddhists who took refuge under the Triple Gem, we have to often remind ourselves that we are disciples of the Gautama Buddha. We've got to develop our potential of seeing the life in the way that the Gautama Buddha taught us. The nature of how merits and demerits construct within one should be comprehended. A sincere effort should be made to prevent accumulating defilements. Most importantly one should be truthful to oneself with a great effort. May you be able to realize the Four Noble Truths. With Metta...

I think your advice is very good for asians, but not appropriate for Western practicioners.

You maybe right @theDoctor. Also the advice given is for the conventional. If you look at suttas, there is no clear grammatical structure. It is the sound that gives the meaning and most verses have “double meanings”: There is an apparently simple meaning, but deeper meanings are hidden most times. In the conventional sense, “atta” means “a person”. The deeper meaning of “atta” is “in full control” or “the essence” or “the truth that is timeless”. I will write another answer – for the ‘Western’ mindset. Do give me a bit of time.

Technically each Dharma will finally lead you to enlightenment.
Start with the one that is easy for you, that you are happy to practice, that does not annoy other people, that you can afford to do.
Similar as you eat, pick up the food that is available for you, that you are happy to eat, that does not cause other being suffering, that you can afford to get.
Such as your name is Richard, you might start to think what is Rich, what does money do with my life, is it important to my life, and keep asking questions until you have answered many questions related to.
Such as your name is Santiago, you might start to ask what is ago, what is time, what has happened to me yesterday, what am I currently doing, what I m going to do tomorrow, do I remember what I did one year ago, what is happened to me when I was young, what happened to me before I was born, and keep asking questions until you are clear of many things related to.
There are lots of Dharma that Buddha has taught during his time. Pick one book and read, and practice according to what the dharma book.
May you be happy and enlightened soon.

I'll try to give a logical answer (I'm worried that a ten-word answer would be easy to misunderstand).
Beware almost every word in the following answer is part of a technical/specialist vocabulary, and each word is the subject (topic) of innumerable "dharma talks" or essays: including words like enlightenment, suffering, cessation, ignorance, etc. They look like English words, but they're translations into English. The original words may have specific meaning and/or a range of meaning, e.g. "suffering" might be translated as "unsatisfactoriness" instead.
What is enlightenment?
Maybe start by understanding what 'enlightenment' is: what's the goal? There are maybe two definitions of enlightenment:

Cessation of suffering (and/or cessation of craving)
Cessation of ignorance

What is the path?
Canonically (according to the Buddha's first and most famous speech) the "path which leads to the cessation of suffering" is what's called the Noble Eightfold Path or the Threefold Training.
What of meditation?
There are at least two or three types of meditation:

Intended to concentrate or pacify the mind (Samadhi)
Intended to see phenomena clearly and learn wisdom (Vipassanā)
Intended to increase 'divine' attitudes e.g. good-will and equanimity (Bhavana and Brahmavihara)

Each type has different ways in which it can be practiced, i.e. there are various explanations (instructions) of techniques for how to practice them ... this answer is meant to be "in short" so I won't try to reference them here.
Note that meditation might be only one or two of the parts of the "Threefold Training".
What more about enlightenment?
Examples of suffering states or non-enlightened states are sometimes called afflictive emotions. There are long lists of them (for example this list ... and there are also lists of factors conducive to enlightenment).
It may be your experiencing these (afflictive emotions) which you'd equate with "suffering" and a desire for enlightenment might be a desire to not experience these.
Assuming you may experience them there are things to know about them that may make them easier to cope with during meditation:

They're impermanent (they come and go), don't try to cling to them
They're part of a cycle of cause and effect
People (minds) are more-or-less susceptible to them, have more-or-less-well-developed mental immunity/habits. Some Buddhist traditions see enlightenment as gradual, a progression; perhaps at an early stage of enlightenment people still experience suffering, but more-or-less cope with it.
They tend to have an immediate cause (e.g. a cause like, seeing something, deciding that you like it, deciding that it ought to be yours and that you want it), and an immediate effect
The Buddha identified some root causes: e.g. craving, attachment, ignorance.

The first factor of the Noble Eightfold Path is "Right View". Developing right view is important, and (conversely) wrong view is a cause of suffering (or more accurately, doesn't lead to the cessation of suffering).
For example if you hold the view that "he hit me, he hurt me" then that's a view associated with suffering. It's wrong, not necessarily in the sense that it's untrue but in the sense that it's unwise or unskillful (to attach to this sort of view if you want to stop suffering -- therefore don't hold to that view).
Some examples of right view include, don't see something as permanent if it's impermanent; don't attach to self-centred views.
To wrap this up, one of the root causes of suffering is craving. One of the types of craving might be a craving for enlightenment! One of the views of enlightenment is that it's a state in which craving (which leads to suffering) has been abandoned ... so for example there's a sutta like this one: Brahmana Sutta. Another view is that it's a state in which you have removed the fuels (causes) of craving.
I recommend this answer because it points out (accurately, I think) that different schools of Buddhism have slightly different takes on what the problem is and what the solution is.
This answer is still short: it doesn't mention teachers, good (and bad) friendship, social responsibility, harmlessness as a guiding principle, other guiding principles such as timelessness or immediacy.

For you, the path to enlightenment begins right here.  Because that's where you put your consciousness, in a Q/A site called "stackexchange".
"Is this the right path" is irrelevant, because you already started by offering a piece of your own lifeforce (time, chi, whatever) into the arena.
So now the universe answers back.  The only question is "do you hear it" (awreness)?  Aim for the highest point.  It could be GOD, it could be Nature or mysticism, or it could be some virtue (Truth, Justice, etc.).  When you aim for the highest point, you will either fail quickly and learn within your lifetime or achieve greatness.
You have to understand what enlightenment is.  On the one hand it means "to be aware" or PRESENT, on the other hand it means your burden is lighter -- because you working your karma -- paying back to those who were wronged before you.
So, there are always two trajectories until you are enlightened:  going upward (seeking enlightenment) or going downward (addressing your karma) so you can go further upwards.
Be careful that your goal of enlightenment isn't burdened with subconscious quasi-goals such as being better than someone else (ego-driven).  That is a common pitfall among Western Buddhists.
As one Buddhist master remarked "after enlightenment, the mountains are mountains" -- don't get wrapped up in the goal, the journey along the way is part of the path.

There is a truth in what you said @theDoctor. Your answer has got me thinking and the need to write another answer to the OP. If you look at suttas, there is no clear grammatical structure. It is the sound that gives the meaning and most verses have “double meanings”: There is an apparently simple meaning, but deeper meanings are hidden most times. In the conventional sense, “attā” means “a person”. The deeper meaning of “atta” is “in full control” or “the essence” or “the truth that is timeless”.

@SapthaVisuddhi: In Jewish study, they recognize four levels of interpretation of scripture called Pardes (Jewish exegesis)

I edited your comment to fix the hyperlink. See here (which I formatted as [See here](https://buddhism.stackexchange.com/editing-help#links)) for a description of how to create hyperlinks on this site (personally I always use the "inline link" style rather than "reference-style link").

For me, the path to enlightenment begins and ends with the Four Right Efforts:

"[i] non-arising of bad, harmful, unhealthy mindstates that have not yet arisen.
  "[ii] abandonment of bad, harmful, unhealthy mindstates that have arisen.
  "[iii] arising of healthy mindstates that have not yet arisen.
  "[iv] maintenance, increase, and development of healthy mindstates that have arisen."

This practice involves classifying all your mindstates in two types: healthy and unhealthy (harmful).

Why don't I divide my thinking into two sorts? Thinking imbued with desire for pleasure, thinking imbued with negativity, [and all other kinds of] harmful thinking - into one sort, -- and thinking imbued with renunciation, thinking imbued with non-negativity, [and all other kinds of] healthy thinking -  into another sort.

The mindstates are harmful when they

...lead to my own affliction or to the affliction of others or to the affliction of both. They obstruct discernment, promote vexation, ...  degradation, and defilement.

While the healthy mindstates are the ones that:

... lead neither to my own affliction, nor to the affliction of others, nor to the affliction of both...If I were to think & ponder in line with that even for a night... even for a day... even for a day & night, I do not envision any danger that would come from it.

And now the practice is very simple:

Whenever harmful thinking had arisen, I simply abandoned it, dispelled it, wiped it out of existence.

So, you should practice this during the day, and when you meditate you practice the same. Rid your mind of harmful thoughts and mindstates, gladden the mind, and make it stable.

In Tibetan traditions, especially in the old Kadampa and the Gelug tradition, there is a genre of teachings called "Gradual Path to Enlightenment" (Lam Rim).
It shows how relying on a qualified teacher matters and how to do so, how to meditate on death and impermanence, abide by ethical discipline, renounce the pleasure of this life, and so forth.
Atisha's Lamp on the path to enlightenment is a short Lam Rim that His Holiness the Dalaï-Lama Commented. Je Tsongkhapa wrote a short, middle-length and great Lam Rim. Here you find his short Lam Rim, The foundation of all good qualities, with Geshe Sopa's commentary.
All in all, my main advise to you: study with a qualified teacher. An I am not saying this teacher ought to be a Tibetan one, as long as he is qualified. Too often do we see people who did not study properly, who have wrong views (new-age like views or Hindu-like views at best) and who think they can understand the texts on their own. They end up thinking that "all is one", that "nothing exists whatsoever", that "this is all a dream". This is neither correct nor healthy. Focus on that rather than meditation. Dharma practice has to be ordinary, you should not hope for extra-ordinary experiences and that would be a risk if you engaged in meditation without having studied thoroughly beforehand.

And Buddha studied with a teacher? :)

Yes he did in the sense that he followed instructions from teachers. The Pali canon shows how he reached the first, then the second, then the third jhana and so forth following teachers' instructions.

And now, we heave the Buddha's teachings to study and practice :)

@Dhammadhatu I recommend Atisha and so forth because although they are free from the fault of fabrication (i.e. they do not invent a teaching but merely explain the meaning of the buddha's teachings) they might be easier to access than, for instance, the Lankavatara Sutra. Furthermore, your statement "Was the Buddha such a bad teacher that he needed others to explain his error that being an arahant is not full enlightenment" is not accurate from the Mahayana viewpoint, since the Buddha himself taught so in Mahayana sutras.

@Dhammadhatu By fabrication, I do not mean 'thinking', but making things up. It is the way Shantideva uses it. Instead of assuming I meant 'thinking' with 'fabrication' and reacting on the basis of an incorrect assumption, you should have asked me.

I have to explain because you did not study the Mahayana teachings extensively and with a qualified teacher. If "language cannot be used in a rigid manner" then the word I used should not matter that much, but you say I could have chosen a better word. There is contradiction. In addition, I did not come up with 'fabrication'. For instance, Gyltsab Je says "b. Rejecting the fault [of this material being of Shantideva’s] own fabrication."

I have never stated that the teacher is more important a factor than the student. Neither have I dismissed teachers of traditions that are not mine ( admire Bhikkhu Bodhi, Bhikkhu Anālayo, etc.) nor would I see every Tibetan teacher as being so qualified. I thought it went without saying, but apparently not. I do not know what you read in speech for you to state that I'm dogmatic, but my power is limited as to the way you take what is given to you (again I agree with your saying the student matters, the taker matters).

It is your right to not admire them. But, admiration aside, you will agree that Dharma practice enables us to nonetheless respect those we do not agree with.

The Buddha was very clear on this in the Satipatthana Sutta (The Foundations of Mindfulness):

This is the only way, monks, for the purification of beings, for the overcoming of sorrow and lamentation, for the destruction of suffering and grief, for reaching the right path, for the attainment of Nibbana, namely, the four foundations of mindfulness.

"This is the only way, monks" is a translation of the Pali

EKYĀNO AYAṀ BHIKKHAVE MAGGO

There are different interpretations to what is meant by this:

There is no other way leading to nibbana (= "Awakening") than Satipatthana
This way must be practiced by oneself alone
This way is not crooked, it's the direct/shortest way to nibbana

At the end of the teaching he says:

"Now, if anyone would develop these four frames of reference in this way for seven years, one of two fruits can be expected for him: either gnosis right here & now, or — if there be any remnant of clinging/sustenance — non-return.
"Let alone seven years. If anyone would develop these four frames of reference in this way for six years... five... four... three... two years... one year... seven months... six months... five... four... three... two months... one month... half a month, one of two fruits can be expected for him: either gnosis right here & now, or — if there be any remnant of clinging/sustenance — non-return.
"Let alone half a month. If anyone would develop these four frames of reference in this way for seven days, one of two fruits can be expected for him: either gnosis right here & now, or — if there be any remnant of clinging/sustenance — non-return.

So practicing Satipatthana would be the best option for you. For a basic understanding of the practice you can read the original teachings itself, and read/listen to explanations by other teachers like these

Ajahn Sumedho: Talk on mindfulness
Bhikkhu Bodhi: Talk on the Noble Eightfold Path
"How To Meditate" by Ven. Yuttadhammo

Of course the only thing that really matters is your actual practice of meditation and mindfulness.

This is the wrong place to start, in the scriptures the sutras begin with saying who the audience were, to indicate the level of difficulty and the prior skills and experience required, for a beginner to try to practice something meant for monks will be very difficult. You need to start with more basic foundational practices.

@Yinxu I don't know what you mean by 'something meant for monks'?

"Thus have I heard. At one time the Blessed One was living among the Kurus, at Kammasadamma, a market town of the Kuru people. There the Blessed One addressed the bhikkhu thus: "Monks," and they replied to him, "Venerable Sir." The Blessed One spoke as follows:"
Usuallly the suttas begin with the audience, if it is a lay person the Buddha will give a simpler lecture on morality or basic practice. He only talk about difficult to grasp concepts to the advanced monks.

http://www.accesstoinsight.org/lib/study/merit.html
"Of all the concepts central to Buddhism, merit (puñña) is one of the least known and least appreciated in the West. This is perhaps because the pursuit of merit seems to be a lowly practice, focused on getting and "selfing," whereas higher Buddhist practice focuses on letting go, particularly of any sense of self. Because we in the West often feel pressed for time, we don't want to waste our time on lowly practices, and instead want to go straight to the higher levels."

"Yet the Buddha repeatedly warns that the higher levels cannot be practiced in a stable manner unless they develop on a strong foundation. The pursuit of merit provides that foundation. To paraphrase a modern Buddhist psychologist, one cannot wisely let go of one's sense of self until one has developed a wise sense of self. The pursuit of merit is the Buddhist way to develop a wise sense of self."

@Yinxu Have you read the whole teaching? (The Satipatthana Sutta)

@Yinxu You could work that into an answer of your own: i.e. answer that the path starts with "the pursuit of merit", and say how (and maybe why) ... basing the answer on references and/or personal experience.

I would only offer two suggestions - find yourself a teacher and empty yourself of expectation.  The first will guide you and the second will lessen the likelihood that you will be misled.  The Buddhist path calls for trust.  It is a forfeiture of belief and preconception done for the realization of true faith.  Close your eyes, fall back, and let the hands of the patriarchs catch you.

The patriarchs are for a different people.  The gurus of old don't know our Western world.  It's origin story is different, it's destiny, and probably even it`s species.

Absolutely, wholeheartedly disagree.  :-)

Why don't you ask your guru?

Afraid I don't have one.  I practice Rinzai; my teacher is American.

The path to enlightenment begins with Right Thought. Right Thought is part of the Noble 8 fold path which is the answer to both of your questions.

Right thought is not common buddhist etymology, to my understanding. Could you elaborate on which path you're referring to?

Right understanding might be better than right thought

@Erik If it "begins with" right thought, I guess that "right thought" is either a reference to "right view", or to sila (the ethical components of the threefold training), or maybe it's intention.

@ChrisW Yes, the OP did clarify this. The reason i asked was because to me, right thinking could mean right view, right mindfulness or right concentration for instance. Perhaps even right effort, et c.

People here come up with all sorts of answers about enlightenment, believing in this or that. I hate to burst your bubble. They are not telling you about enlightenment because they don't know what it is. If nobody knows what the hell it is, how can they tell you how to achieve it? They only have concepts of something they read, which they have not verified.
Buddha really meant to reduce the suffering we are currently undergoing in our lives by techniques like meditation. How do you feel after meditating? If done properly, you will feel calm right? And then later you would be back, your villain mind will be back. This mind always looks after itself. It's ALWAYS LOOKING OUT FOR "YOU". So when this mind asks for "enlightenment" it is trying to add onto itself! "Become better" or something like that. It's looking for something more and then something more, endlessly. It's like wanting to be a billionaire.
We always desire this or that and think about the future or past. Be clear about this. Buddha did NOT talk about enlightenment. I repeat there IS NO ENLIGHTENMENT in the way you or your mind thinks there is. If you think so, you are unnecessarily bullshitting yourself and will chase your own tail and end up nowhere. It's not outside you. But if you clear yourself out, I hope you find it, God willing or yourself willing ;)
Gautama Buddha is only telling you about reducing your level of mental misery. The very fact that you are searching for something means you are not okay with whatever you have currently.
I'll only tell you one thing here. By looking at the enlightenment of other people, I think they mean self-annihilation or self-destruction. Your concept of your own self. Destroy that and see yourself as a part of everything in this universe. When I say "you" cannot achieve enlightenment, I mean this current thing which you think is "you". Enlightenment is when you somehow find out you are NOTHING but a blob of energy which will leave its current form when the universe needs it. That's it. Everything else is, I am sorry to use this word, bullshit.
Now I don't know how to do this clearly, because my thoughts get in the way everytime telling me this or that. That's all the karma. A memory of myself. I don't want to get into the details.
That's why I said you have to clearly know what you are looking for. If you don't know it, don't bother with it.

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Goddess in Buddhism
When I did bit research, I found that. All the Goddess in Buddhism are Demons (Yesha) by origin. After getting enlightenment and following from Buddha's teaching, they are transform from Demon to Goddess.
For Example: Hārītī Ajima who use to kill and feed on human children, become the goddess to protect infant children (Very Interesting Story).
I am just interested to know more about Buddhism. I heard there are certain steps or processor that need to achieve to reach such Divine stage.
So just wondering what are those process or steps called and there definition??
If possible can, someone can share sites with interesting and motivation stories on Buddhism??

Each Dharma book is an interesting story :D

This Dharma book contains many little stories: http://read.goodweb.cn/news/news_more.asp?lm2=2385 (However I did not read the English version)

@SIDU Thank you, That site looks awesome, Please share if you know More links

I read Chinese versions :) So I do not know much English website, however I read many Chinese versions on the next site, not sure if you can find any English versions: http://www.cttbusa.org/

Found: http://www.cttbusa.org/sutratexts.asp

Before all I appreciate your interesting towards the Buddhism my friend. I have few clarifications to be done before answering.
First thing you have mentioned "All the Goddess in Buddhism are Demons" is completely wrong. Ladies who were died with good and powerful enough karma wipaka to be born in heaven will born as goddess.
Actually there are very few english translations are available about dhamma.
I recommend you to read "wimana wattu and Pretha watthu". there are lot of stories contain information about people who were born among heaven and hell.

In Tibetan Buddhism, a deity is an Enlightened person in my experience.

It is a very important question that you asked IMO.  It is because we live in very unfortunate times. The outside forces and other distractions are so many that it is extremely difficult to practice the Dhamma in this lifetime. So it is vital that we make sure that we are born in the heavenly worlds amongst the Devas.
Realizing the fact that there is not going to be another chance of getting the protection of the Dhamma in this human world, we should cultivate the Sēkha Bala Dhamma – Saddhā (faith), Sīla (virtue), Sutha (Dhamma knowledge), Thyāga (generosity), and Paññā (wisdom of the Dhamma) – at this present time. These are qualities that are found in the Devas. If you develop them, then there is divinity within you.
If a person gets a chance to be born among divine beings by developing the Sēkha Bala Dhamma in this life, he/she will get another chance to practice the Dhamma. But that chance will be missed if that person gets a birth in the human world or a lesser world instead. That is because we do not know how long this chance of practicing the Dhamma will last in this human world.
Therefore, you should realize the importance of getting a help from this life by wisely cultivating wisdom and mindfulness. If you get a birth in the Thuthisa Divine World for instance by realizing this importance and cultivating the Sēkha Bala Dhamma, then you will definitely get the protection to your life. Scriptures talk about many Deva worlds.  So you can get the chance to complete the rest of the path of the Dhamma even in the Chatummahārājika Divine World. If you get a birth either in Thāvathinsa, Yāma, Thusitha, Nimmānharathi, or Paranimmitha Vasavatthi Divine World, you will still have the protection in the Dhamma since you already have the Sēkha Bala Dhamma developed in you. The teachings of the Supreme Buddha clearly show this protection – it is your duty to secure it to your life.

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Questions about cetasikas
1) How are cetasikas expressed? Could they be expressed as how the mind is orientated, thus influencing how, when, and what cittas arise?
2) Do cetasikas therefore play any indirect role in the patticasamuppada process (dependent origination, did I spell the Pali right), say, for example, by governing what vēdana arises in response to contact?

I think I understand the second question, "Where do cetasikas fit into the doctrine of the 12 nidanas?" So the first question, "How are they expressed?", means something like "What is their effect, consequence, manifestation?".

Before getting to those questions, it's important to know exactly what the term cetasika mean. See Ven. Bodhi's excellent "Comprehensive Manual of Abhidhamma"'s Chapter 2 at:https://store.pariyatti.org/Comprehensive-Manual-of-Abhidhamma-A--PDF-eBook_p_4362.html

I have personally never studied 'cetasikas' but have studied Paticcasamuppada in detail.
Paticcasamuppada explains the 12 conditions (paccaya) rather than causes (hetu) that condition the origination or generation of psychological suffering (sorrow, lamentation, pain, grief & despair).
Therefore, it is important to distinguish between 'generic vedana' arising at contact (such as vedana experienced by a fully enlightened arahant that has attained Nibbana) and 'ignorant vedana' arising in paticcasamuppada, which occurs due to ignorance.
Since 'contact' is a 'cetasika' (per Wikipedia), all vedana is a response to the citasika of contact.
However, in paticcasamuppada, every one of the 12 conditions (and their sub-constituents, which total at least 54) arise due to the cetasika of ignorance/wrong-view/delusion, as described in the quote below:

To an uninstructed, run-of-the-mill person, touched by feeling that which is felt born of contact with ignorance, craving arises. That fabrication of 'self'
  is born of that. And that fabrication is inconstant, fabricated,
  dependently co-arisen. That craving... That feeling... That contact...
  That ignorance is inconstant, fabricated, dependently co-arisen.
Parileyyaka Sutta

1) Cétasika can be expressed in general as Vedanä, Sangna, Sankhara.

Vedana - Experience
Sangna - Recognition
Sankhara - Constructing  Formations (pancha upädänakkhanda)

Cétasika is considered an ultimatum in meaning. The other three ultimatum are Citta, Rupa and Nibbana. Cétasika are broken down in detail in Abhidhamma.
2) Cétasika is very detailed and occur in each conscious moment. Hence their contribution to Paticcha Samuppada is involved in many stages. Avijja, for example is backed by cétasika. Sankhara is backed by cétasika. Vinñana is essentially Citta and Citta don't arise without accompanying cétasika. Nama Rupa is essentially pancha upädänakkhanda, which in turn is Rupa, Vedana, Sagna, Sankhara, Vinñana.
It all gets messy if, one tries to mix them up without fully understanding them individually (understanding pancha upädänakkhanda, understanding paticca samuppada). Once you understand them well, it's important to refer to Abhidhamma before trying to correlate them.

Just because you have specifically asked as to what “Cetasika” means I will give a brief introduction. But I have no interest in giving the sources for it, as to where I took it from. It is because I do not want to see any of my Dhamma friends to go into this level of detail. Life is fleeting. It doesn't last as long as we would like. We have precious little time to practice this Dhamma. We cannot even make sense of the simpler discourses in the Sutta Pitaka, so why would one want to go into such?. The Sutta Pitaka consists chiefly of discourses delivered by the Buddha Himself on various occasions. In it the mental factors are beautifully explained, and one need NOT go any further.
“Cetasika” is found in the Abhidhamma. In my opinion it is there for the “I think therefore I am” type – the so called deep thinkers, who are too smart for their own good. Anyway this is what “Cetasika” means:
There are 52 cetasikas (mental factors), which, arising together in various combination, giving rise to what we call the Mind. Of these fifty-two, Vedana and Sanna are treated as two distinct groups. The remaining fifty are collectively called Sankhara.  Unlike the Conventional Teachings of the Suttas, the later day  intellectualization  of the Dhamma came up with  Abhidhamma or  Paramattha Desana  (An Ultimate Doctrine). This Abhidhamma Pitaka is composed of the following works:-- Dhamma-Sangani (Enumeration of Phenomena); Vibhanga (The Book of the Treatises); Katha Vatthu (Point of Controversy); Puggala Pannatti (Description of Individuals); Dhatu Katha (Discussion with reference to Elements); Yamaka (The Book of Pairs); Patthana (The Book of Relations).
Four ultimate things (Paramattha) are explained in order in the Abhidhamma. They are Citta, (Consciousness), Cetasika (Mental concomitants), Rupa (Matter) and Nibbana. The “being” is microscopically analysed and its component parts are minutely described in it.
A citta is a single act of consciousness - "a state of mind." Each citta consists of many components. Consciousness - the basic experiencing of the object is taken seperately. Together with consciousness every citta contains cetasikas or mental factors.
The cetasikas is said to color the citta and give it its distinctive character.  By reference to cetasikas, citta is broken down to: a mind that is lustful, a mind without lust, a mind with or without aversion, a mind with delusion or without delusion, a cramped or scattered mind, a developed or undeveloped mind, a surpassable or unsurpassable mind, a concentrated or unconcentrated mind, the freed or unfreed mind. Mind is not identified with as "I" or "mine," not taken as a self or as something belonging to a self. it may be in a pure state or a defiled state, a lofty state or a low state. Likewise this analytical doctrine, this analysis goes on and on and on minutely describing everything. If that is what you want you can start reading the Abhidhamma. Good luck to you!

In my experience with studying the Theravadin Abhidhamma, if found that the 52 cetasikas did not make much sense until I studied the citta-vithi (thought process). Then it made more sense how or why one citta (moment of consciousness) is different from another in terms of the cetasikas.

In Dhammasangini for different cittas the possible lists of chetasikas are given.  These lists contain much more than 52 chetasikas.  The 52 chetasikas have been obtained by removing chetasikas of similar meaning and including chetasikas from other sources. (like Anupada sutta) Some chetasikas as seen above give a 'color' to chittas (kusala, akusala).  Some are doing a definite action (phassa, sanna, chetana etc)

Cetasikas can be put into 3 main division: feeling, perception and fabrications. It is the mental proliferation and volitional activities which create the fabrications. This might be the key elements in trying to piece together how it related to Dependent Origination.

The meeting of the three is contact.
With contact as condition, there is feeling.
What one feels, one perceives.
What one perceives, one thinks about.
What one thinks about, one mentally proliferates.

Madhu,piṇḍika Sutta
Craving and clinging closely related to I and Mine conceiving (thinking about and mentally proliferating) as there is a subject who want or do not wants. [Tanhā Jālinī Sutta] Me and mine conceiving is through "thinking about" (from above list). [Sevitabbâsevitabba Sutta] So thinking about and mentally proliferation is related to craving and clinging also.

with the six sense-bases as condition, there is contact;
with contact as condition, there is feeling;
with feeling as condition, there is craving;
with craving as condition, there is clinging;

(Paṭicca,samuppada) Vibhanga Sutta

There other interesting relationships also. Here again thinking follows perception.

dependent on the diversity of elements, there arises the diversity of perceptions
dependent on the diversity of perceptions, there arises the diversity of thoughts
dependent on the diversity of thoughts, there arises the diversity of desires
dependent on the diversity of desires, there arises the diversity of passions
dependent on the diversity of passions, there arises the diversity of searching?

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text:
The Four Nutriments: what do the metaphors in SN 12.63 mean?
In the Puttamansa Sutta, the Four Nutriments ('ahara') of physical food, sense contact, intention & consciousness are discussed using four comparative similes ('metaphors').
The 1st simile of eating the flesh of one's own son (merely for the sake of crossing the desert) is very straightforward. However, the remaining three similes of a 'skinned cow', of 'two strong men dragging a man into a pit of fire' & of a 'thief punished by a king' are not obvious in their meaning.
These three similes are:

And how, O monks, should the nutriment sense-impression be considered? Suppose, O monks, there is a skinned cow that stands close
  to a wall, then the creatures living in the wall will nibble at the
  cow; and if the skinned cow stands near a tree, then the creatures
  living in the tree will nibble at it; if it stands in the water, the
  creatures living in the water will nibble at it; if it stands in the
  open air, the creatures living in the air will nibble at it. Wherever
  that skinned cow stands, the creatures living there will nibble at
  it.
And how, O monks, should the nutriment volitional thought be considered? Suppose, O monks, there is a pit of glowing embers, filled
  to cover a man's height, with embers glowing without flames and smoke.
  Now a man comes that way, who loves life and does not wish to die, who
  wishes for happiness and detests suffering. Then two strong men would
  seize both his arms and drag him to the pit of glowing embers. Then, O
  monks, far away from it would recoil that man's will, far away from it
  his longing, far away his inclination. And why? Because the man knows:
  'If I fall into that pit of glowing embers, I shall meet death or
  deadly pain.'
And how, O monks, should the nutriment consciousness be considered? Suppose, O monks, people have seized a criminal, a robber, and brought
  him before the king saying: 'This is a criminal, a robber, O Majesty!
  Mete out to him the punishment you think fit!' Then the king would
  tell them: 'Go, and in the morning strike this man with a hundred
  spears!' And they strike him in the morning with a hundred spears. At
  noon the king would ask his men: 'How is that man?' — 'He is still
  alive, Your Majesty.' — 'Then go and strike him again at noontime with
  a hundred spears!' So they did, and in the evening the king asks them
  again: 'How is that man?' — 'He is still alive.' — 'Then go and in the
  evening strike him again with a hundred spears!' And so they did.

What do we think each of these similes means?

Detailed explanations of the metaphors are available at http://dharmafarer.org/wordpress/wp-content/uploads/2009/12/20.6-Puttamamsa-S-s12.63-piya.pdf

@santa100 Sorry but could you re-post that as an answer? Though that might be a good answer but, if it's posted as a comment the OP can't "accept" it (to finalize this topic).

@santa100 I was asking for our own opinion however I did read the link. Apart from Piya Tan's personal explanation of wholesome & unwholesome volition (which sounds reasonable to me), I am not satisfied the link explains the metaphors. For me, the explanations are very difficult to follow, almost impossible to understand, particularly Nyanaponika's explanations. The idea of 'rebirth-consciousness' does not exist in the suttas therefore this cannot be an explanation of a sutta. Regards

@Dhammadhatu I'm not sure exactly what part of the explanation from prof. Tan or Ven. Nyanaponika make it impossible for you to understand? I don't think the idea of brushing aside concepts that are not part of the Sutta Pitaka a useful approach. Afterall, they didn't call it the "Tipitaka" for no reason (the 3 Baskets of Discipline, Suttas, and Commentary).

Nyanaponika refers to ordinary people seeking sensual pleasure & willingly accepting pain in pursuit of such pleasure. I find this does not explain the metaphor since the metaphor is addressed to bhikkhus. As for Piya Tan, what he writes is not part of the Sutta Pitaka since Piya Tan is obviously not the Buddha but merely a lay scholar & commentator, the same as we are. As I posted, the metaphor is not explained in the sutta thus I thought it worthwhile we consider for ourselves what it means. Piya Tan is offering his opinion & inteprertation similar to we ourselves can. Regards

I have read Ven. Nyanaponika's booklet 'Nutriments of Life'. For me, it is a very negative view of the nutriments & the world & not related to the sutta. The sutta states physical food is a nutriment that must be consumed with moderation & blamelessness. The sutta does not state physical food is evil & must be completely shunned. I imagine the same attitude is applicable to the other three nutriments & the metaphors are meant to portray a similar meaning to the simile of the 'son's flesh'. Regards

I am happy to see this quiz or riddle being attempted. For many years I used to wonder about this teaching of nutriments, was never satisfied with  Ven. Nyanaponika's explanation & could not work out a satisfactory explanation for myself.
My personal interpretation is as follows:

The skinned cow refers to all sense contact, including the sense contact of arahants, which still experience pleasant & unpleasant vedana (feeling). Therefore, the noble disciple comprehends sense contact affected by feeling (vedana) will always occur like the skinned cow & thus stops/eradicts craving & attachment towards the 'creatures' ('vedana') nibbling at sense contact.
The two strong mean are good & bad intentions, which both can lead to trouble & suffering, thus the English saying: "The road to hell is paved with good intentions". Therefore, the noble disciple acts with wise intention rather than good or bad intention.
The 'thief' or 'criminal' is attachment (upadana), which takes possession of ('appropriates') sense objects of sense consciousness as "I", "me" & "mine". The Buddha taught that the sense spheres of consciousness are "not-yours". Therefore, the noble disciple does not, like a thief, take possession of sense objects that do not belong to them (otherwise, they will be punished by dukkha each time they take possession of a sense object).

The eye is not yours: let go of it. Your letting go of it will be for your long-term happiness & benefit. Forms are not yours...
  Eye-consciousness is not yours... Eye-contact is not yours... Whatever
  arises in dependence on eye-contact, experienced either as pleasure,
  as pain, or as neither-pleasure-nor-pain, that too is not yours: let
  go of it. Your letting go of it will be for your long-term happiness &
  benefit.
"The ear is not yours: let go of it...
"The nose is not yours: let go of it...
"The tongue is not yours: let go of it...
"The body is not yours: let go of it...
"The intellect is not yours: let go of it. Your letting go of it will be for your long-term happiness & benefit. Ideas are not yours...
  Intellect-consciousness is not yours... Intellect-contact is not
  yours... Whatever arises in dependence on intellect-contact,
  experienced either as pleasure, as pain, or as
  neither-pleasure-nor-pain, that too is not yours: let go of it. Your
  letting go of it will be for your long-term happiness & benefit.
Na
  Tumhaka Sutta: Not Yours

The Four Nutriments ('ahara') of physical food, sense contact, intention & consciousness are true for the worldlings who have craving (tanha) and greed (lobha) or aversion / revulsion (dosa), caused by ignorance (moha).  Tanha, lobha, dosa, moha provides nourishment for the continuation of life from existence to existence.
The worldlings have  five aggregates of clinging (Panca Upadana Khandas).  In the case of Arahants – the awakened ones - there is no craving. They have rooted out the “I concept”.  They have fully realized five groups of existence and the five grasping groups of existence (Panncakkhandha and panca upadanakkhandha) .  Arahants have only the five aggregates in detachment  (Anupadana).  Upadana means ‘grasping’ and Anupadana means ‘non-grasping’ or letting go.While Upadana leads to Samsara, Anupadana leads to Nibbana.
To put it in another way, (for better understanding), after the elimination of all defilements an arahant lives in society without Upadana or Attachment to any thing in the world. He does not have the five grasping groups (pancaupadanakkhandha) but only five groups,aggregates (Pancakkhandha), namely, rupa (body), vedana (feeling), sanna (perception), sankhara (volitional formation), and vinnana (consciousness).  His consciousness is ‘translucent’ (for wont of a better word) as no food is provided. In this he is immeasurable - is like the Buddha. All measurement is from a here to a there, with in-betweens. In the Buddha, there is neither a here nor a there nor any in-between. He is thus immeasurable. See Bahiya Sutta: About Bahiya - Ud 1.10

“Then, Bahiya, you should train yourself thus: In reference to the seen, there will be only the seen. In reference to the heard, only the heard. In reference to the sensed, only the sensed. In reference to the cognized, only the cognized. That is how you should train yourself. When for you there will be only the seen in reference to the seen, only the heard in reference to the heard, only the sensed in reference to the sensed, only the cognized in reference to the cognized, then, Bahiya, there’s no you in that. When there’s no you in that, there’s no you there. When there’s no you there, you are neither here nor yonder nor between the two. This, just this, is the end of stress.” (Ud I.l0)...

An arahant is sustained by ONLY the physical food that does not create upadana. He does not have the other three ‘mental’ foods as he has rooted out the “I concept”.  He is free from attavada upadana by seeing it as a deceit (the true nature). So if there is no nutriment  consciousness, an Arahants  consciousness is mere presence.  Since there cannot be presence without something being present, it is what distinguishes living from non-living, such as tables and chairs. The Buddha has stated that consciousness is dependently arisen, since without a condition there is no origination of consciousness. In an Arahant, Paticcasamuppada (dependent arising) ceases.  Which means Vinnana paccaya (conditioned by consciousness) too ceases.

While the sutta did not state the nutriments are evil, it did give vivid and explicit similes of the son's flesh, the skinned cow, the man being dragged toward a burning pit, and a criminal being pierced with three hundred spears. Needless to say, one doesn't have to be very smart to think carefully before indulging oneself with the nutriments after hearing what the Buddha just said. Imho, SN 12.63 reflects the Dukkha inherent in all conditioned phenomena. And the Four Nutriments are no exception. For us regular worldlings, we tend to only understand the 1st of the 3 aspects of Dukkha, Dukkha-dukkha, for the obvious reason that it's right there in front of our naked eye. That's why we continue to indulge in the nutriments and sense pleasures. It would take a significant degree of cultivation and penetrative insight to understand the other 2: Viparinama-dukkha and Sankhara-dukkha. And until then, we'd still only be able to have some superficial understanding of what SN 12.63 was trying to say. Per ChrisW's suggestion, I've reposted as an answer and include the 2 very helpful links from Prof. Tan and Ven. Nyanaponika with detailed commentary on SN 12.63:
http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/20.6-Puttamamsa-S-s12.63-piya.pdf
http://www.accesstoinsight.org/lib/authors/nyanaponika/wheel105.html

Existence is nourished by four nutriments as pointed out in the Sutta in @Dhammadhatu ‘s OP. They are essential to continue life in this repeated existence in samsara. This is true even to the deities in the celestial world and brahmas in the Brahma world. Of these four kinds, one is material/physical, and the other three are mental nutriments. The first is the easy one as @Dhammadhatu pointed out. But how can one see one’s mental state or mental growth if not for the guidance that the Buddha provided in many of the suttas? The Buddha provided the explanation in the Atthi Raga Sutta (where there is passion) in the Samyutta Nikaya.
How we have to view the four similes is that ONLY if we eradicate the Cause, would the Result be eradicated, and NOT otherwise. It is our Desire for Life that prevents this. The day that we establish wisdom over ignorance, and liberation over desire, we will see the light. The day we remove our passion, delight and craving for the four nutriments Consciousness will not arise. Then mentality-materiality won’t happen, and as a result there is no growth of fabrications, and no renewed becoming. If no future birth, there is no decay, death, sorrow, lamentation, pain, grief and despair.
To explain the 2nd simile -  that of the nutriment sense-impression:
Sense impression (phassa ahara) is Contact.  So what is contact? It is the coming together of three things. It is the eye, the visible object and consciousness coming in contact with each other (seeing forms). It is the ear, sounds and consciousness coming in contact with each other (hearing sound). it is the nose, odours and consciousness coming in contact with each other (smelling odours), and likewise. What is important to understand is with the arising of consciousness, it is inevitable for contact to arise. Only if we eradicate the desire for what we see, hear, smell, taste, touch and think, will the contact and consciousness not arise. The simile of the flayed cow is about this contact that one cannot get away from as long as there is desire.
To explain the 3rd simile -  that of the nutriment volitional thought:
Volitional thought (Cetana) is Intellectual Intention. It is the arising of wholesome or unwholesome intentions. As before, these mental factors are a result of desire. Beings who cannot comprehend that these mental factors are not permanent, that they constantly change, and be mindful, gets thrown into the pit whether they like it or not. Only getaway is eradicating the cause – the arising of ‘cetana’ – of intellectual intention.
To explain the 4th simile -  that of the nutriment of Consciousness:
 Our sense faculties are constantly contacted and stimulated because of Consciousness as explained before. The two that I did not touch on are that of the body and intellect.  Dependent on the body and tactile sensations, arises consciousness at the body.  Dependent on the intellect (ideas/thoughts), there arises consciousness at the intellect. With the Consciousness, desire arises. This arising of passion, delight and craving results in greed, hatred and delusion – these are the spears that one gets shot at with. As long as you have passion, delight and craving, you are alive and get shot at and experiences pain and distress.
So our Desire for the Dhamma, to live in the Dhamma should be even greater than our desire for Life. Those are the ones who will stay in the finger tip. What the Buddha has said is patisotagami or against the current of conventional thinking. Buddha knew this well. That is why He said,

“Dear Bhikkhus, ones who would not realize this Dhamma is like this earth (in amount). Ones who realize this is like this small amount of soil on my finger nail.”

I found your answer thoughtful however I have gained the impression the answer is  saying that a Buddha has no arising of consciousness, a Buddha has eradicated intention, that desire is the cause of sense contact (thus an arahant has no sense contact since an arahant has no desire) & desire arises with each arising of consciousness. Although the similes in the sutta are not overt & difficult, I am not satisfied your answer explains them despite your excellent effort.

@Dhammadhatu, it is been a while since I posted a response to your question, but no one else has come up with another answer. Maybe I will wait a few more hours before replying to your comment. It will not be my opinion & interpretation, but as stated in other suttas of the Buddha. I know next to nothing. It is only the Buddha who knows, and it is all in the suttas. It is because of the Abhidhamma, later additions to Kuddhaka Nikaya, and other 'Adhamma's that we have complicated things. We really do live in unfortunate times when the adhammas are more powerful.

Thanks. I only raised the question because I used to be very curious about this teaching since I was unable to make sense of it. Yes, the nutrients are taught in different ways in the suttas, including removing craving towards them. Removing craving is obviously the right thing but it is the metaphors I am curious about. Regards.

Its simple, i will explain each 4 nutriement in points
(1) physical food- it is to be regarded only for survival ,not for flavour or intoxication.  We should treat our physical food like we are feeding on our young baby
(2) contact- if contact happens with our six senses (eye, ear,  skin ,etc) to its corresponding object (vision, touch etc)  then that object chew on it.. for  example if you have eye and lets say its corresponding object is pornography.. then we should regard pornography as something chewing our eye.. ultimately destroying it...but since we normally dont question our sense-object , we enjoy it till our ruin.
(3) volition- buddha said you should regard volition as survival instinct only.... here a man hates death.. but a practitioner sholuld not fear death... so he is saying normally we have volition that act upon survival instinct..we dont act truly but act according to survival instinct
(4) consciousness- well , if u are ever depressed, tensed and wanted to end your life ,,you should know its all because of your damn consciousness...so consciousness is spear acting upon us..buddha is saying to take this consciousness in command
Finally,  what he meant is that the above 4 food is necessary for survival. . If you cant survive then how will you attain nirvana...but he is saying dont take these food unmindfully..see what game is undergoing behind the scene..in 1 word right view ...right view about these food is necessary for salvation ..hope it helps. If any question please ask. Glory to Buddha

Son's flesh - eating itself is unsatisfactory (as the sensations produced by it is unsatisfactory - Kīta,giri Sutta). eating should be only to maintain life to fulfil holy life and not for the please of it.
Skinned cow - there is not place or way to avoid contact. With contact follows sensations [Paticca-samuppada-vibhanga Sutta, see Dependent Origination] which are unsatisfactory [Cūla Vedalla Sutta], hence it nibbles.
Pit of glowing embers - all volitions are accompanied by sensations throughout the body. [121 mental states] Whatever sensations end in misery, thus burns. Also this is the fuels which drives you and burns you until death
Robber's struck with spears - it is thought consciousness that you know and experience. If not for consciousness you will not feel sensations. [Dhātu Vibhaṅga Sutta] Being alive is like having consciousness. All sensation are unsatisfactory [Cūla Vedalla Sutta], hence the pain of been struck by spears.

If not for consciousness, there is no enlightenment. From MN 43: "Discernment (wisdom) & consciousness are conjoined, friend, not disjoined. It's not possible, having separated them one from the other, to delineate the difference between them. For what one discerns, that one cognizes. What one cognizes, that one discerns." The views of  Nyanaponika Thera were unbalanced, overly negative & not in accord with the purpose of the teaching of the four nutriments.  Regards

Here's my personal interpretation of SN 12.63.
To understand each of the similes, the clue is in the epilogue of each simile. The nutriments are not taken literally. Each simile becomes harder to interpret than the previous one. Each simile is also related in some way to the previous simile.
First simile
The epilogue says "When the nutriment edible food is fully understood, lust for the five cords of sensual pleasure is fully understood." The five cords of sensual pleasure (MN13) are forms, sounds, aroma, flavours and tactile sensations.
The nutriment of edible food here is not literally edible food, but refers to the external sources of forms, sounds, aroma, flavours and tactile sensations.
The simile teaches us that these "edible food" should be used only as a means to get through samsara, while living the holy life, to reach the end of suffering, and one should not have passion for them.
The nutriment of the external sources of the five cords of sensual pleasure, nourish and sustain the five cords of sensual pleasure. Literal edible food is the source of flavours, for example.
Second simile
The epilogue says "When the nutriment contact is fully understood, the three kinds of feeling are fully understood." The three types of feeling (Iti 52) are pleasure, pain and neither-pleasure-nor-pain. There are six types of contacts (SN 12.2) - eye-contact, ear-contact, nose-contact, tongue-contact, body-contact and intellect-contact.
The cow is exposed to different environments like wall, tree, water, open air. These are like contacts. The creatures biting her are the three types of feeling. The biting itself is craving. The painful sensation that arises from the biting is suffering (from dependent origination: from feeling as a requisite condition comes craving; and from the second noble truth: the cause of suffering is craving).
The simile teaches us to abandon passion for the three types of feeling arising from contacts. The creatures (feeling) are fine, but don't let them bite you (craving). For the bite is painful (suffering). The nutriment of contacts nourish and sustain the three types of feeling.
To link back to the previous simile, SN 12.44 says for example, "dependent on the ear & sounds there arises ear-consciousness. The meeting of the three is contact". So, five types of contacts are directly related to the five cords of sensual pleasure. The sixth type of contact is indirectly related to the five cords of sensual pleasure.
Third simile
The epilogue says "When the nutriment mental volition is fully understood, the three kinds of craving are fully understood." The three types of craving (Iti 58) are craving for sensuality, craving for becoming and craving for non-becoming.
The word for "nutriment of mental volition" in this sutta is manosañcetanāhāro. From SN 12.38, we read: "what one intends, and what one plans, and whatever one has a tendency towards: this becomes a basis for the maintenance of consciousness." I interpret mental volition here as intention (cetana) and latent tendencies (anusaya).
SN 12.38 also says "If one does not intend, and one does not plan, but one still has a tendency towards something, this becomes a basis for the maintenance of consciousness", which shows latent tendencies are required for the maintenance of consciousness, even if intention isn't present. There are 7 types of latent tendencies in AN 7.11.
The two strong men are either latent tendencies (anusaya) or latent tendencies and intention (cetana). They grab the man and drag him towards the charcoal pit with glowing coals. The glowing coals are craving. If the man comes into contact with the glowing coals, the arising pain is suffering (from the second noble truth: the cause of suffering is craving).
The simile teaches us to abandon latent tendencies. Don't let the strong men (latent tendencies) drag you into the charcoal pit of glowing coals (craving). Run away from the strong men (latent tendencies). The coals are painful (suffering). The nutriment of latent tendencies nourish and sustain the three types of craving.
To link back to the previous simile, the three types of feeling will not result in the arising of craving, if there are no latent tendencies.
Fourth simile
The epilogue says "When the nutriment consciousness is fully understood, name-and-form is fully understood." Based on the definition of name-and-form in SN 12.2, name-and-form is a kind of mind-body link (according to Theravada interpretation).
And what is consciousness? From dependent origination: from consciousness as a requisite condition comes name-and-form. From the traditional Theravada interpretation of consciousness in dependent origination, this is the consciousness that results in the continuity of suffering beyond physical death, called "rebirth linking consciousness" (cuti citta) in the Abhidhamma.
This is supported by SN 12.39: "When consciousness is established and has come to growth, there is a descent of name-and-form", and SN 12.38: "When consciousness is established and has come to growth, there is the production of future renewed existence." When the suttas talk about the growth, increase, maturing, landing and establishing of consciousness, it essentially talks about consciousness resulting in the descent of name-and-form ("rebirth").
In dependent origination, craving leads to clinging, which leads to becoming. These show the outflow of effluents or mental fermentations or taints (asava), which according to this "cyclic" question causes the arising of ignorance. Then ignorance in turn causes the arising of effluents once again (MN 9), which is mental formations (sankhara) including latent tendencies (anusaya). This then causes the arising of consciousness, which causes the descent of name-and-form (mind-body link) i.e. "rebirth".
This loop back from craving-clinging-becoming back to ignorance is hinted at, in SN 44.9: "when a being sets this body aside and is not yet reborn in another body, I designate it as craving-sustained, for craving is its sustenance at that time."
The bandit or criminal is the one who delights in craving, the fool from SN 12.19, who clings (steals). The king could be ignorance or samsara. The king's men who strike him with spears are mental formations (especially latent tendencies). The spears are (rebirth linking) consciousness. The striking of the spears is the descent of name-and-form (that is "rebirth"). The pain caused by the striking of spears is suffering (birth, aging, death, sorrow, lamentation, pain, distress and despair). Morning, afternoon and evening could be different aeons.
Wouldn't three hundred lifetimes be painful? Well, even one lifetime is painful enough, don't you find from experience? The simile teaches us to abandon stealing (craving), in order to not get struck ("rebirth") by spears (renewed "rebirth linking consciousness") again.
This is reinforced in the  Tears Sutta (SN 15.3):

From an inconstruable beginning comes transmigration. A beginning
  point is not evident, though beings hindered by ignorance and fettered
  by craving are transmigrating & wandering on. Long have you thus
  experienced stress, experienced pain, experienced loss, swelling the
  cemeteries — enough to become disenchanted with all fabricated things,
  enough to become dispassionate, enough to be released."

The nutriment of (rebirth linking) consciousness nourishes and sustains the (descent of) name-and-form.
To link back to the previous simile, the three types of craving (plus clinging and becoming) are the effluents that cause ignorance, and ignorance causes effluents (latent tendencies especially) in turn, that causes (rebirth linking) consciousness to rise again, causing the descent of name-and-form ("rebirth").
Case of arahants
How does this apply to the case of arahants?
From Thanissaro Bhikkhu's commentary of Iti 44:

With fuel remaining (sa-upadisesa) and with no fuel remaining
  (anupadisesa): The analogy here is to a fire. In the first case, the
  flames are out, but the embers are still glowing. In the second, the
  fire is so thoroughly out that the embers have grown cold. The "fuel"
  here is the five aggregates. While the arahant is
  still alive, he/she still experiences the five aggregates, but they do
  not burn with the fires of passion, aversion, or delusion. When the
  arahant passes away, there is no longer any experience of aggregates
  here or anywhere else.

A living arahant experiences the nutriment of edible food and the nutriment of contacts, but he has ended the nutriment of mental volition (i.e. latent tendencies) and therefore also ended the nutriment of (rebirth linking) consciousness. Then, consciousness has no place to land or get established any more (SN 12.64) i.e. no more descent of name-and-form (SN 12.39) meaning no more continuity of suffering after physical death. After parinibbana, all nutriments would have ceased permanently.
Conclusion
The sutta is talking about dependent origination, what keeps dependent origination going, how to end dependent origination, and the truth of the continuity of suffering beyond physical death.
It is true then that the noble eightfold path does not just lead to the cessation of (craving for) form, but also the cessation of (the rearising of) form. (see this question)

Very good effort but too all over the place for me, particularly the superstition about three hundred lifetimes being painful. Lol.... the Buddha taught every time the mind grasps something as "I", "me" & "mine", this is painful; this is the "thief" getting punished each time the grasp or steal. Life without grasping is not painful.

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text:
How should somebody who has developed samadhi verify that rebirth occurs after death?
I am trying to understand how somebody following the dhamma is supposed to verify what the Buddha said about rebirth. In the past I've been told that once somebody has developed samadhi/calmed the mind to a great extent then they will be able to see that rebirth occurs after death.
Usually samadhi helps along the path by making it easier to observe thing and see the world as it is. It's clear how this helps to see anatta/anicca/dukkha, but I do not understand how samadhi helps with seeing anything related to rebirth. I don't know what anyone would begin to observe to see rebirth.
This question has a very similar title to my previous question but it is not a duplicate. In the previous question I had one idea on how someone could verify rebirth occurs after death but the answers said that my idea was incorrect. This new question is asking for other ideas on how someone can verify rebirth through experience.

Go into a very deep samadhi. Very deep.
When in that very deep samadhi, make a subtle mental formation arise. Observe.
Notice how with the arising of the mental formation, awareness comes to be. That very same awareness is consciousness. Within it is the subtle sense that feels the mental formation.
Let the mental formation cease by going back to deep samadhi.
Notice how with the ceasing of the mental formation, awareness goes away. That very same going away of the awareness is death. Within it is the absence of the subtle sense that feels the mental formation.
The way to get additional insight about this is to go into a very deep samadhi and arise a VERY pleasant mental formation. Observe.
Notice how with the arising of the VERY pleasant mental formation, awareness comes to be. Within it is the attachment/craving for that pleasant mental formation. As long as craving/attachment is present, the mental formation is there and the feeling is pleasant and awareness is present. That very same pleasure is "being alive".
Now, don't let the VERY pleasant mental formation to cease, but crave for it to maintain it. Crave for it! Want it! Sooner or later the pleasant mental formation will cease by itself. When it ceases, as long as there is craving/attachment for that pleasant mental formation, the feeling is unpleasant and awareness is present. That very same unpleasant feeling is "fear of death". The cessation of that very same unpleasant feeling is "death".
Now, don't let the craving/attachment for that pleasant mental formation to cease, but crave for it! Really crave for it! Want it! Sooner or later the pleasant mental formation will arise by itself. When it arises, as long as there is craving/attachment, there is also the pleasant mental formation and the feeling is pleasant and awareness is present. This very same pleasure is "being alive".
The wisdom you'll gain by experiencing the above is that awareness consists of many little formations arising and ceasing with high frequency, thus making it seem that consciousness is steady and pleasant. In individuals who don't practice the dhamma, there are so many of these formations, that even if you shut off all your body senses (eye, nose, tongue, ear, body, mind), they would still be arising and ceasing. These formations can be dependent on your body or independent of the body. Impossible is to know exactly. They can come from deep within you. It's like a spider web that has no ends. It's these formations that condition other formations until the center of the web is created: your physical body.
With the death of the body awareness ceases similar to going into a deep samadhi, and craving/attachment "fear of death" is present ... this very same craving/attachment responds/reacts to the many little formations, thus creating new formations and rises the awareness because of the many new formations until a new body is created.
This must be experienced. It's the only way to verify that rebirth occurs after death.
Once you get the above insight, it's simple to follow a logical conclusion:
By wishing/acting/deciding/doing we're doing activities that create formations.
By activities we create subtle senses and come into existence. Then, because of our further activities, we create the senses of our body (eye, nose, tongue, ear, body, mind) and we're born.
By gaining knowledge we are changing our current activities and creating new activites.
When we're changing our current activities and creating new ones, we're also changing our first activities that we acted upon that propelled us into this existence.
With the dead of the body our activities on the surface (eye, nose, tongue, ear, body, mind) are destroyed. And what remains? The primary activities that are changed.
Because these primary activities are changed, after death we will come into a new existence that will be changed too.
Thus, there is no death. There is just the changing of awareness from one existence to the other due to activities. This process is constantly happening ... even now in this very same moment. It will happen as long as craving/attachment is present.
Then, the logical way to continue the path towards the ultimate would be: When craving/attachment stops, all activities stop and suffering stops. Thus, the next step is to stop craving/attachment.

I think this is a fascinating way of recreating death. I don't think I should expect an answer which shows that the physical brain isn't necessary for awareness.

I dont practise Samatha meditation but afaik you can only verify it either by gaining insight and understanding of moment to moment rising and ceasing of reality or by remembering past lives by developing appropriate Samatha meditation.

It is very important to distinguish between full samadhi and upacara-samadhi (sometimes called neighbourhood samadhi, which is true but also misleading). Upacara-samadhi leads to insight or knowledge because it allows one to form concepts that place the object of awareness into a frame of reference or understanding. This cannot happen in full samadhi because of the one-pointedness of mind removes any context in which the object can be understood. It is for this reason that (in the Vissudhimagga, for example) upacara-samadhi is more important than full samadhi in Buddhist literature. You encounter reincarnation when you happen to recall a previous lifetime. A more formal practice is to remember previous events backwards in time. When you do this, you do not get involved in individual recollections1. With practice, the sequence of recollections becomes something like watching a flat stone skip over water. Eventually, the stone makes an additional skip into your previous life. This may be just the feelings you had before death. You may have to catch several skips before you get perspective on your previous life.

1 By "you do not get involved in individual recollection", I mean that When dealing with unwholesome sankhara in upacara-samadhi (vipassana), it is important to recall the formation of sankhara, one must trace backwards in time its development until you reach the memory of the experience that caused a maladaptive decision (karma). This gives you a chance to revise that decision based upon more recent experience. In this way, unwholesome sankhara can be undone. In this context, an individual recollection is generally not helpful.
In order to understand an individual recollection, it must be placed in a context. The recollections that arise from upacara-samadhi are from another dimension of reality that is difficult to understand. In order to understand karma, for example, you must recall a sequence of events backwards in time that are linked together in a special way. Another problem is that the unconscious is far more intelligent than the conscious mind, making it difficult to understand. Generally, an individual recollection requires a sophisticated explanation.
When saying "another dimension", I am talking about the realm of "mind-forms" that I write about in my book, The Path of Love. This the vast system of chakras and nadis that form the mind-base and the psychic body. My experience of this dimension resulted from my exploration of the Theravadin Abhidharma. Mind-forms are perceived by means of the siddhi called anima, which unfortunately is very rare. All intelligent consciousness is based upon the citta-vithi described in the Abhidharma. Stephen Phillips referred to anima as micro-psi perception. Through the micro-psi perception of atoms, quantum physics will be transformed into a deterministic theory. These studies are far beyond the work needed to achieve Enlightenment. They are the science of the remote future. I mention it only in order to answer the (implied) questions in the comments below.

what do you mean by "you do not get involved in individual recollection"?

When dealing with unwholesome sankhara in upacara-samadhi (vipassana), it is important to recall the formation of sankhara, one must trace backwards in time its development until you reach the memory of the experience that caused a maladaptive decision (karma). This gives you a chance to revise that decision based upon more recent experience. In this way, unwholesome sankhara can be undone. In this context, an individual recollection is generally not helpful.

Thank you, your explanation makes total sense, although the phrase "individual recollection" still hangs there with no context. Why "individual"? Recollection of what?

In order to understand an individual recollection, it must be placed in a context. The recollections that arise from upacara-samadhi are from another dimension of reality that is difficult to understand. In order to understand karma, for example, you must recall a sequence of events backwards in time that are linked together in a special way. Another problem is that the unconscious is far more intelligent than the conscious mind, making it difficult to understand. Generally, an individual recollection requires a sophisticated explanation.

Not a bit clearer, to be honest. The only another dimension I have experienced first-hand was a dimension of alternative realities within the quantum ambiguity of the metaverse. But it seems you're talking about some other another dimension.

@AndreiVolkov Ronald Cowen posted an answer to your comment, which I have merged into the original answer above.


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Is there a budhistic way to experience suppressed emotions?
See question above. So far I know to kickstart emotion : writing about emotion, boxing, put your awareness to bodily feelings(kind a hard if you just feel numbness)

Some people think this question isn't clear. I think this question might be answerable as-is, but if you'd like to add further information a few questions that come to my mind are: are there any particular/specific emotions you're thinking of? What is a "suppressed" emotion? Does it exist, can it be experienced, if it's suppressed? Why do you think there is a Buddhist way to experience suppressed emotions, why or to what extent might a Buddhist (or the Buddha) experience (or want to experience) such a thing? What problem are you trying to solve?

I sense a good question hidden in an unclear framework. Try to increase precision and elaborate on your understanding of surpressed emotions.

@ChrisW "are there any particular/specific emotions you're thinking of?" Not particularly, maybe I just wanna be more emotional in general, I am super stoic. I don't think it's healthy to be that attached. Most of the happiest people I know have reach emotional lives, they cry heartfully, same for laughing and raging. "What is a "suppressed" emotion?" Maybe they don't exist, however people can go catatonic where they shut off. I think suppressed emotions are the same, people just shut themselves off because pain is unbearable.

@ChrisW "Why do you think there is a Buddhist way to experience suppressed emotions" they seem to be pretty advanced in psychological stuff "why or to what extent might a Buddhist (or the Buddha) experience (or want to experience) such a thing?" I read somwhere that experiences masters have laugh wholeheartely, Cry intensly etc. and that there are far from beeing robotic. I think they want to experience it because of the old no happiness without crying paradigma. What problem are you trying to solve? "What problem are you trying to solve?" beeing robotic, numb, detached, low energy/motivation.

By prolonged meditation, suppressed emotions & experiences will generally inevitably rise into to the surface of consciousness awareness.

This is a very very important question. I have practiced mindfulness meditation for 50 years. I have been a novice monk for 30 years (only recently has it become possible to get full ordination in Canada and now I do not need it). First of all, a definition of emotion: An emotion addresses an issue or situation of overriding importance. It is unconsciously motivated. It causes the mind to address the issue or situation with energy and focus. It gives the highest priority or importance to dealing with the issue or situation. That is why it can be hard to hold back. But, if the emotion is very painful or sad, then you will suppress it as best you can, but it can still make you very unhappy. In Buddhist terminology, this state of unhappiness (or any emotion for that matter) is called "kamma-vipaka," which merely means that, in some previous decision making process (citta-vithi), you decided on the importance of the issue. Since being loving or being loved is such a profoundly important issue in life, you are understandably subject to relentless suffering if you have come to the conclusion that you are not lovable. Even though everyone is born in an Enlightened state and therefore very loving and lovable, subsequent experiences (such as those experienced in a dysfunctional or abusive family) can cause you to decide that you are unlovable. This is a profoundly tragic situation. The problem is that you often forget making that tragic decision so that you are stuck with the unhappiness it causes. Fortunately, the Satipatthana-Sutta provides a way to recall (re-experience) that original decision. First of all, it is advisable to practice mindfulness of breathing for awhile. Mindfulness of breathing consists of paying attention to the physical sensation of the breath at the nose. When a distraction arises (which is perfectly natural and is not a mistake), you return your attention to your breathing so that you do not get caught up by the distraction. The purpose of this meditation is to maintain a state of objectivity (so that your Bodhicitta gets a chance unconsciously correct misunderstandings). In a more advanced state of mindfulness you pay attention to "the body in the body," which happens to be the "felt sense" as described by Eugene Gendlin in his book called "Focusing." (I suggest you study this book.) By recalling the original decision, you have the opportunity to correct it in terms of subsequent experience. You have a liberating "Aw-ah" experience.


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How does extinction fit in the 12 links?
I'm thinking that the presence itself, in sense objects, of ignorance, the first link, is never erased; that with final nirvana something has being, the absence of the 12 links. Can the absence of causation itself be present like sense objects are, in absolutely any Buddhist philosophy?
Or is that silly, like saying that the there exists the non existence of a crow's teeth?

If you want to learn more about existence/non-existence you should read about non duality where buddhist philosophy dives into the idea of neither existence nor non-existence.

The ideas of existence (atthitañceva) & non-existence (natthitañca) are both 'self-views' although often not translated as such (per SN 12.15) but sometimes translated as such (per SN 12.17, Iti 49, etc). 'Existence' means "I exist and will continue to exist' and 'non-existence' means "I do not exist & I will cease to exist". Both contain "I" belief. Neither existence nor non-existence merely means no self-view (per SN 12.15). Neither existence (atthitañceva) & non-existence is about physical things.

@Dhammadhatu so there's no "existence" without "self existence"?

There is but that is not really the subject of suffering & liberation but can be interesting philosophy. Regards

In Buddhism, there are many kinds of causation. See this link about the five 'niyama' or 'laws of causation' (physical, biological, mental, moral & dhamma/spiritual).
The twelve links are the conditions leading to suffering; the cessation of which leads to Nirvana. They are 'dhamma niyama' or 'spiritual causation'. The twelve links are not mental causation (citta niyama).
Therefore, sense objects can be experienced without ignorance because mere sense objects (which are mental causation) are not suffering in themselves.
The only absence of causation is Nirvana, when the twelve links stop creating suffering.
As for the other four kinds of causation, they keep rolling on (unless human life becomes extinct, which would make extinct moral causation & leave three laws of causation).

When you ponder, weigh, and compare the teachings of the Buddha, you have to base it on adopting the right attitude and asking the right questions about them. AN 2.25 points out that some of the teachings are meant to have their meaning inferred, whereas others are not. Only a Stream Entrant will be able to see this difference. So you and I have a way to go.
If everything is impermanent and transitory, how can there be anything be present or absolute?  Because all things are thus conditioned and transient (anicca), they have no real independent identity (anatta) so do not truly ‘exist’, though to ordinary deluded minds this appears to be the case. Simply stated Dependent Origination can be said as,
With this as condition, That arises. With this NOT as condition, That does NOT arise.
This "Coming to be, and Ceasing to be" is happening all the time. All phenomena are subject to this unending interaction. And since all phenomena are dependent on other phenomena, they are all transient and impermanent. Avidyā the first condition.

Can the absence of causation itself be present like sense objects are, in absolutely any Buddhist philosophy?

There is only one this arisen by casue - Nirvana. Everything else is asien due to a case. Sense objects, sense faculties and contact and the sensation that follows are also arise by cause. Sense faculties cease when you are in Nirvana.

with final nirvana something has being, the absence of the 12 links.

Nirvana is when you break out of the vicious cycle so it cannot be part of the dependent origination.

I'm thinking that the presence itself, in sense objects, of ignorance, the first link, is never erased;

As long as you do not understand the 4 Noble Truths. You have Ignorance. This asies at the point of contact. When you contact an object you feel a sensation and perceive the object not inline with its true nature and react with craving or aversion. How to break Ignorance at this point is to be aware of the true nature (arising and passing) of the experience keeping you mind firmly equanimous.


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the Pair of Knowing and Object
I am searching for a precise understanding of what Buddhist authors mean by the "pair" of "knowing" and "object". For example

...we see that what we call "self" is simply the pairwise progression of knowing and object.

A second example

In all perception, at each event of noting, there is always this twin
  pair, the object and the mind,  which observes the object. These two
  elements of the object and the knowing mind always arise  in pairs,
  and apart from these 2, there does not exist any other thing in the
  form of a 'person',  an 'observer', an 'experiencer', nor any 'Ego',
  'Subject', or  any 'Self'! No identity is present! Similarly is there
  no 'observed object', or  any 'substance' "out there" independent of
  the mind!

Can this "pair" of "knowing" and "object" be explained precisely? Or is it subject to many interpretations?
What insight is gained from discussion of this pair of knowing and objects?

You speak of an instance of a phrase, what phrase are you referring to? I have not read the word "pair" in any traditional text I have studied, where did you find it? What do you mean with "identifying this pair"? Maybe you could clarify your question a little.

The "pair" ("dyad") exists in a number of texts: http://www.accesstoinsight.org/lib/authors/nanananda/wheel183.html#passage-21

The "pair" or "dyad" of "knowing" and "object" is found in the Pali scriptures, as follows:

At Savatthī. “Bhikkhus... there is
  this (internal) 'group' ('kaya': of five aggregates) and external minds-and-bodies: thus this dyad (pair). Dependent on the
  dyad (pair) there is contact.
https://suttacentral.net/en/sn12.19

~~~

Owing to a dyad (pair), monks, consciousness comes into being. And how, monks, does consciousness come into being owing to a dyad?
Owing to the eye and forms arises eye-consciousness. The eye is impermanent, changing, 'becoming-otherwise.' Forms are impermanent,
  changing, 'becoming-otherwise.' Thus this dyad is fleeting and
  transient; impermanent, changing and 'becoming-otherwise.'
Owing to the ear and sounds arises ear-consciousness. The ear is impermanent...
Owing to the nose and scents arises nose-consciousness. The nose is impermanent...
Owing to the tongue and savors arises tongue-consciousness...The tongue is impermanent.
Owing to the body and tangibles arises body-consciousness. The body is impermanent...
Owing to the mind and ideas arises mind-consciousness. The mind is impermanent...contacted, monks, one feels. Contacted, one intends.
  Contacted, one perceives. Thus these states also are fleeting and
  transient; impermanent, changing, 'becoming-otherwise.'
SN 35.93

~~

The six internal media should be known. The six external media should be known. The six classes of consciousness should be
  known.
MN 148

The 1st example in the opening post is incorrect understanding. For example, what is called "self" is not included in the pairwise occurring of knowing and object. The mind can know an object but be empty of 'self' because 'self' is a product of thought/thinking (sankhara) while knowing an object is a product of sense consciousness (awareness: vinnana).
The 2nd example in opening post is correct understanding, in that when the dyad operates, there is no 'self' in its operation. However, the thought of 'self' can arise quickly afterwards (called 'becoming'), as described in the following quote:

"Who, O Lord, has a sense-impression?"
"The question is not correct," said the Exalted One.
"I do not say that 'he has a sense-impression.'... The sixfold sense-base is a condition of sense-impression and sense-impression is
  the condition of feeling.'
"Who, O Lord, feels?"
"The question is not correct," said the Exalted One.... And to that the correct reply is: 'sense-impression is the condition of feeling;
  and feeling is the condition of craving.'"
"Who, O Lord, craves?"
"The question is not correct," said the Exalted One. "I do not say that 'he craves.'...the correct reply is: 'Feeling is the condition of
  craving, and craving is the condition of clinging.'"
"Who, O Lord, clings?"
"The question is not correct," said the Exalted One, "I do not say that 'he clings.'... 'Craving is the condition of clinging; and
  clinging is the condition of the process of (self) becoming.' Such
  is the origin of this entire mass of suffering.
SN 12.12

However, to assert there are no objects "out there" independent of the mind goes too far. It is an extreme view because to assert there are no objects independent of mind asserts the mind is 'God' or 'Brahma'; that the mind is the 'Creator' of the external world.
The Pali scriptures state, for example, the Laws of Nature exist "out there" regardless of whether they are known by a human mind or not, as follows:

Monks, whether or not there is the arising of Tathagatas, this property stands — this steadfastness of the Dhamma, this orderliness
  of the Dhamma: All processes are inconstant. All processes are
  unsatisfactory. All phenomena are not-self. The Tathagata directly
  awakens to that, breaks through to that. Directly awakening & breaking
  through to that, he declares it, teaches it, describes it, sets it
  forth. He reveals it, explains it, & makes it plain
Dhamma-niyama Sutta

The insight gained in knowing this pair of knowing and objects is explained in SN 35.93 quoted above, namely: (i) the mind can clearly identify or be aware of them and not create 'self-becoming' & suffering in relation to them; (ii) the mind can see they arise & pass; are impermanent & not-self.
For example, an object of sight comes to be when eye-consciousness functions & an object of sight ceases to be when the eye-consciousness ceases to function. If this is watched, the mind will realise all sense objects are impermanent each time the eye blinks.

Joseph Goldstein has at least several references to the pair of knowing and object in his book...if not a Buddhist teaching, then why do I see this in so many places?

That mind & external objects are a 'pair' is a Buddhist teaching but the quotes you provided are not Buddhist teachings but, instead, misinterpretations of Buddhist teachings. You need to read my answer very carefully because the distinctions I have made are very subtle.

I have done so. Thank you. What are the origins of the "pair" : knowing and objects? i.e. why did Buddha introduce this, to impart us with what insight?

The purpose of the pair is for many reasons: (i) so the mind can clearly identify or be aware of them and not create 'self-becoming' & suffering in relation to them; (ii) so the mind can see they arise & pass; are impermanent & not-self. An object of sight comes to be when eye-consciousness functions & an object of sight ceases to be when the eye-consciousness ceases to function. If this is watched, the mind will realise everything is impermanent each time the eye blinks. Thus, the mind will not cling to anything in the world. But to assert there are no objects "out there" is going too far.

I will rewrite my answer in about 20 minutes to make it clearer (when I finish doing my work). I am currently doing two things at once on two computers.

I misread the 2nd example. It is correct. I will have to change my post again. This comes from doing two tasks at the same time.

You do not have to worry yourself. Your answer had the correct intention.

Its OK. I am finished now. I enjoy posting dhamma. It is never a worry.

You seem to be asking about the relationships among the concepts of "self," "action," and the "object" or situation targeted by that action. (In Buddhism, the perception of an object is an action.) You could say that an agent or self threw (or saw) a ball. Or you could say that a set of beliefs and wishes in the form of an action-plan caused the ball to fly through the air (or caused the perception of the ball). The first is a form of object-causality. The second is a form of event-causality. In event-causality, the agent or self or subject disappears. Confusion arises when a person believes that talk about object-causality implies that, when switching to event-causality, one should become aware of the self as an object of awareness. The real problem arises when, in some kind of meditation, a person imagines an experience of a self and slips into full samadhi and believes he has achieved greatness.