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Is rebirth essential to Buddhist philosophy?
Elements of Buddhism can be rationally accepted such as the suffering that arises through attachment, the benefits of meditation, and even the acceptance of anatta, or non-self. It seems, however, that a belief in rebirth cannot be accepted rationally. A believer must suspend rationality to accept a theory that describes the migration of some level of consciousness across a lifetime. A common analogy used is the light emitting from a candle. Nothing substantial transmigrates, however something passes over from one life to the next. This theory cannot be directly perceived. Therefore, it cannot be tested, or verified rationally. It is simply a metaphysical inference.
Buddhism encourages debate and questioning of its theories, and to not accept everything on face value. How then can a Buddhist test this theory? Can one disregard this element of Buddhism? Is it an essential aspect of the philosophy?

This topic is a near-duplicate of (and so you may also find answer at) Is rebirth a delusional belief?

Possible duplicate of Is rebirth/reincarnation central to Buddhism?

I would say it is essential. But while most of the doctrine may be arrived at by metaphysical analysis rebirth seems to be a sort of 'lemma' in that it cannot be proved or falsified in logic. Perhaps this is why it is treated by some as dispensable. Immediate experience seems to be the only way to confirm it.

From Ven. Bodhi's excellent short essay "Dhamma Without Rebirth?":

The aim of the Buddhist path is liberation from suffering, and the Buddha makes it abundantly clear that the suffering from which liberation is needed is the suffering of bondage to samsara, the round of repeated birth and death. To be sure, the Dhamma does have an aspect which is directly visible and personally verifiable. By direct inspection of our own experience we can see that sorrow, tension, fear and grief always arise from our greed, aversion and ignorance, and thus can be eliminated with the removal of those defilements. The importance of this directly visible side of Dhamma practice cannot be underestimated, as it serves to confirm our confidence in the liberating efficacy of the Buddhist path. However, to downplay the doctrine of rebirth and explain the entire import of the Dhamma as the amelioration of mental suffering through enhanced self-awareness is to deprive the Dhamma of those wider perspectives from which it derives its full breadth and profundity. By doing so one seriously risks reducing it in the end to little more than a sophisticated ancient system of humanistic psychotherapy.

If SN 12.2 together with SN 23.2, SN 5.10 & SN 22.81 are studied carefully, it appears what the Buddha actually taught about was "birth" ("jati") rather than "rebirth", which is not physical birth from the womb of a woman but instead the birth of the idea, view or mental concept of "beings" ("satta') or, more simply, "self".
Thus, the goal of Buddhism is to eradicate the birth of the "self" idea, i.e., eradicate attachment,i.e, realise not-self (anatta). The Alagaddupama Sutta states:

There is the case where a monk's conceit 'I am' is abandoned, its root destroyed, made like a palmyra stump, deprived of the conditions
  of development, not destined for future arising.
There is the case where a monk's wandering-on to birth, leading on to further-becoming, is abandoned, its root destroyed, made like a
  palmyra stump, deprived of the conditions of development, not destined
  for future arising.

It is obvious believing in post-mortem "rebirth" is not essential to Buddhism because many fully enlightened beings came into existence after the Buddha spoke his first three sermons and none of those three sermons mentioned "rebirth". Each of those sermons only mentioned the eradication of craving & self-view.
The Pali scriptures explicitly state that the dhamma of the Buddha is visible here-&-now & is to be verified by each wise person thus "rebirth" cannot fall into the dhamma of the Buddha.
The common analogy about the light emitting from a candle is not logical, which is why the Buddha never taught such an analogy. In Buddhism, only a later-day monk famed for teaching Greeks is known for such an analogy. This is because the light of the candle depends on the wick & wax. The light (flame) of a candle cannot move to a second candle if the first candle is devoid of wick & wax. Further, the first candle cannot even be lit without fuel from other source. This is why the Buddha taught consciousness cannot arise without a physical body & without sense organs & admonished a monk for teaching consciousness is reborn (refer to Mahatanhasankhaya Sutta & Upaya Sutta).
Buddhism originally criticized other religions that believed in unverifiable things  (refer to Canki Sutta, Tevijja Sutta, Tittha Sutta, Bhūmija Sutta, etc) but later-day Buddhism over the years must have started to teach & write scriptures about unverifiable things.
The Pali scriptures themselves explain any belief about any kind of "rebirth" is not a factor of the noble eightfold path but is, instead, a polluted right view that sides with only morality (refer to Maha-cattarisaka Sutta).

Rebirth
Rebirth and death is momentarily. Each instance the material composition of your body changes, arising and passing away. Physical death is not different. See this answer, this answer and this answer.
How then can a Buddhist test this theory?
By doing Samatha meditation some people can develop memory of past lives. This way you can ascertain it. Also buy validating what you remember. Also there are studies like: Many Mansions: The Edgar Cayce Story on Reincarnation
Can one disregard this element of Buddhism? Is it an essential aspect of the philosophy?
This is an essential part. Each subatomic particle (Kalapas) arise and pass away. Main practice of Buddhism (vipassana) is to understand this at the experiential level.
And finally use this knowledge and insight at the moment or physical death to release oneself from the cycle of birth and death.

"Elements of Buddhism can be rationally accepted such as the suffering that arises through attachment, the benefits of meditation, and even the acceptance of anatta, or non-self. It seems, however, that a belief in rebirth cannot be accepted rationally."
Indeed, the Buddha described his Dhamma as "The Dhamma visible here & now". However, if that visibility were immediately accessible, we wouldn't have doubts about it, and it would not be necessary a "path" to help us "see it". So, knowledge and skills are necessary to develop understanding, just like knowledge and skills are necessary in sciences to "see" phenomena (e.g. whether the claim of climate change affected by humans is true or not).
In the same way, knowledge and skills are necessary to see past lives according to the Buddha.

"When my concentrated mind was thus purified, bright, unblemished, rid of imperfection, malleable, wieldy, steady, and attained to imperturbability, I directed it to knowledge of the recollection of past abodes. I recollected my manifold past abodes, that is, one birth, two births ... 'There I was so named, of such a clan, with such an appearance' ..."
-- MN 36

"A believer must suspend rationality to accept a theory that describes the migration of some level of consciousness across a lifetime."
One does not need to accept (believe) or reject a theory that one is unable to verify at the moment. One can simply be aware of the competing theories and strive to see which one, if any of them, is true.
In science something similar happens where researchers point out flaws or remaining questions that need to be answer (i.e. that need more research) before accepting a theory.
I find this similar to the way the Buddha taught as well (e.g. refer to the discourse of DN 1). Thus, generally speaking, it's better to try out methods to see the truth than to hold or cling to any view prematurely. Or, more than that:

Who is attached still enters into doctrinal debates,
  but one unattached, how could he take sides?
  For him nothing is taken up or put down,
  With all views shaken off, relying on none.
-- Snp 4.3

Finally, the idea of "what transmigrates" is an old and on going debate, as the Buddha explicitly denied that any substance transmigrates, which would make that very substance something that endures -- that is, a Self (see this question: Then where did the concept of “rebirth” come from?).
"A common analogy used is the light emitting from a candle. Nothing substantial transmigrates, however something passes over from one life to the next. This theory cannot be directly perceived.  Therefore, it cannot be tested, or verified rationally. It is simply a metaphysical inference."
As explained above, the buddhist doctrine asserts this knowledge can be directly perceived without resorting to fragile or indirect means (like beliefs). So, in the context of buddhism, recalling past lives is as metaphysical and irrational as the idea of remembering what one did a few moments ago.
"Buddhism encourages debate and questioning of its theories, and to not accept everything on face value. How then can a Buddhist test this theory"
I'm not able to recall my past lives, so my knowledge is not rooted in practice, only in reading. The progression of meditative attainments the texts shows seem to indicate that one who has mastered the fourth jhana (samadhi practice) possess enough skills in concentration to direct his/her mind to memories of past lives. I've heard that one practice is to gradually recall childhood, infancy, even memories in the womb, and then, to the past life, and so on.
"Can one disregard this element of Buddhism? Is it an essential aspect of the philosophy?"
This question has been asked here: Is rebirth/reincarnation central to Buddhism?.
There is, though, a Secular Buddhism movement in the west which seem to, literaly, strip out a lot of things from Buddhism, including rebirth. Also, some buddhists who may not subscribe to this movement may also believe rebirth (as in dying and reappearing in another life) is not true or not part of what the Buddha taught.

The Pali actually does not state: "he recollects his numerous past lives". This is a mistranslation & misrepresentation of the teachings. Note these following proper translations that use "homes" & "abodes": http://www.accesstoinsight.org/tipitaka/dn/dn.02.0.than.html & https://suttacentral.net/en/sn22.79 Regards.

The SN 12.70: Susima Sutta explains many arahants declared enlightenment without any recourse to recollection of past abodes. http://www.accesstoinsight.org/tipitaka/sn/sn12/sn12.070.than.html

@Dhammadhatu all the translations I know (Walshe, Nanamoli, Bodhi...) use "past lives" in these contexts -- or mean "past lives". It may be the case that a literal option could be adequate (not always the case, as any translator can attest) or that these translators are letting their bias show. However, it's a stretch to say this is misrepresentation.

I posted a translation from Bhikkhu Bodhi that does not use "past lives".

KARMA IS ESSENTIAL TO BUDDHISM. Buddha said that if one doesn't believe in karma then it's wrong view. There are many proof for karma. You just need to look back into your life and remember all the bad things that you did and the similar bad experiences that you suffer.

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I wonder if we should distinguish between rebirth and karma. .

Rebirth is regarded by the Supreme Buddha as essential and foundational to the pursuit of His teachings. This is very evident and is clearly illustrated in the Dependent Origination (Paticca samuppāda) that the being is born in the Becoming (bhava) caused by the grasping or attachment (upādāna) as stated "Upādāna paccaya bhavō, bhava paccaya jati...," But many people have mistaken the Becoming given here to soul or ego (ātma). This clearly shows that 'Rebirth' is integral to the Dependent Origination taught by the Supreme Buddha and is one of the twelve reasons that all of us are unable to comprehend. What are these twelve reasons?

Dependent on ignorance or lack of knowledge arises volitional actions or thoughts or kamma formations: Dependent on volitional actions or thoughts arises consciousness; where the consciousness is created, there exist mind and matter or mental physical phenomena and caused by them, arise spheres of senses (salāyatana) or faculties. Those faculties cause to contact and then get to sensation or feeling. Through feeling arise, desire thirst or craving (tanhā) Through craving arises grasping or clinging which leads to the process of Becoming where the concept of being exists. This existence of the concept of being is subjected to the concept of birth decay, illness and death etc.

One option is to neither believe nor disbelieve the actions of rebirth.
The very definition of belief is rooted firmly in the absence of fact. The same is true for a non-believer. Somewhere between the two lies a spacious void where one can remain open-minded about the notion of rebirth.
Personally, I shelved it precisely in this way. Two years later insight had occurred and not from books or other people. But still, I'm not able to prove it so it's neither here nor there. You'll have to find out for yourself. Wondering about it in this way may become huge hindrance to your practice. There's nothing wrong with entertaining your curiosity though if you keep the seriousness out of it.

How do the "rationalists" (those who try to just depend on what they have already understood) argue how and why they come into being with all their diversity, desires, character... and where is might disapear intom From nothing to nothing? God? Mystic? A mysterious plan?
Is there any valid argument beside of the fact of being reborn and that birth has a cause?
And what is the cause of birth?

Then, in the first watch of the night, he gave close attention to dependent co-arising in forward and reverse order:
From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-&-form. From name-&-form as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging-&-death, sorrow, lamentation, pain, distress, and despair come into play. Such is the origination of this entire mass of suffering & stress.
From the remainderless fading and cessation of ignorance comes the cessation of fabrications. From the cessation of fabrications comes the cessation of consciousness. From the cessation of consciousness comes the cessation of name-&-form. From the cessation of name-&-form comes the cessation of the six sense media. From the cessation of the six sense media comes the cessation of contact. From the cessation of contact comes the cessation of feeling. From the cessation of feeling comes the cessation of craving. From the cessation of craving comes the cessation of clinging/sustenance. From the cessation of clinging/sustenance comes the cessation of becoming. From the cessation of becoming comes the cessation of birth. From the cessation of birth, then aging-&-death, sorrow, lamentation, pain, distress, and despair all cease. Such is the cessation of this entire mass of suffering & stress.

It's all around and within oneself possible to prove rebirth, day by day, moment by moment, and the truth of cessation of becoming can be proved as well. Just cut off not-knowing, ignorance, and the wheel is stopped: release. Real-ease.
[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]

Yes, it is. Without it the central concept of kamma would automatically and directly be baseless and meaningless. I would like to offer a down to earth explanation by means of an example from real life. If I am completely wrong, I will appreciate any downvotes and criticism.
If you go to a party, let's say, for clarity's sake, with a lot of your relatives. Let's say it's a wedding party, so conduct is at least somewhat expected. You might get drunk, let loose, and forgo self criticism. You might put up a show, make stupid jokes, hit on ladies that should not be hit on. If you are lucky, some other folks might do similar things while drunk, but especially if you were the only one to misbehave, then, when you wake up in the morning, and it all starts to come back, then it's that feeling that indicates that you have re-emerged (re-born) at a lower plane as a direct result the uncontrolled stupidity you commited during the night before — everyone saw and knew what you did would cause you suffering, but they didn't tell you because you wouldn't have listened, or they did and you didn't listen, so they waited until this very moment that you wake up alone here and now. I would say that is the phenomenon of telepathy. If now, you exit your bedroom and you still "don't remember", people will give you faces of a certain type which will ensure that you cannot ignore your "rebirth", most likely warm and friendly smiles (if they are good people).
And then you say to yourself: "damnit, karma..." (or, "damnit, kamma...", if you are a devout Pāli purist!). And you better go and apologise, and do something to compensate. As well as clean up the mess of your social suicide, after which you will have time to contemplate on the mess prior to the suicide which led you to it in the first place.
This example is easy to follow simply because it all happens very intensely and during a short period of time, and is made up of events of obvious meaning and ramifications, unlike long term deeds and fruits, the connections of which are harder to detect in everyday life — it's like in particle accelerators such as CERN, where everything that is normally slow (spread out in time and space), is made very intense (brought together in time and space), so as to "un-rare" certain normally rare events. You can generalize from the party and hangover example, just as scientists generalize from very intense high-speed collisions. So if for example a man for a long period in his life ignorantly/irresponsibly/uncompassionately, out of sheer sexual desire, sleeps with young easy to seduce drunk girls at parties, one day he will meet a child of one of such girls, who has had to grow up in a family where such a girl, now a mother, still has wounds or scars or emotional voids from such lifestyle, and the upbringing of that child will have been more difficult or lacking as a result. He/she will look you in the eye and something inside you tells you it was you who contributed to it, "in a previous life". Of course the girl/mother herself is at least as responsible obviously, and she might go through a similar rebirth once her child looks her in the eye after having met with such a man, and possibly having connected the ends at some point later in life.
If a person desn't believe in it — and I'm sure every person has experienced something like this first hand, unless perhaps he is Tarzan or Mowgly — then he obviously doesn't believe in reality, i.e. he is ignorant of it. And the Tathagata always teaches us about the reality.

The concept of karma, at core, is an expression of the cyclical nature of our universe. Day becomes night becomes day; storms alternate with calm, and sunny days with cloudy; seasons swing round and round; creatures are born and give birth to others like them; emotions rise and fade away, only to rise again later. This is part of why the breath is an important focus in many meditations. The breath naturally and inevitably cycles. We can't stop it, but it's not fixed; it shifts and changes as things shift and change around and within us, but it always goes and it always returns. In mathematics this is known as a chaotic orbit, where feedback interdependencies between forces produce unpredictable results with a recognizable pattern of repetition. It is very much a part of our world.
I imagine we can all see how the notion of rebirth would arise from this: if so many other things in the world come and go and come again, why not life itself? But fixating on the idea of rebirth tends to undercut the broader understanding of karma as cyclicality.
The notion of personal rebirth, as I see it, is a comforting simplification, bordering at times on superstition. The egoic self chafes at the idea that there is an end to its existence, so the egoic self imagines that it will go away and return in a different body when this body is done. But of course, Buddhist practice aims at putting this egoic self to rest, and if we let it rest then the idea of rebirth becomes more a matter of mimesis than reincarnation. The attitude we take towards the world is reflected in the world, such that it can be reborn in the world at a different place and time. If we move though the world with rage, we foster a world in which our kind of rage can build itself in others; if we move through the world craving sensual passions, we foster a world in which that desire for sensual passions is engendered in others. If we move through the world without clinging and craving, we foster a world in which we do not influence others to cling or crave, and those attitudes are not reborn.
I'm not saying that there is no personal rebirth. Maybe there is, in some way I don't understand. But if there is such a thing, it is unimportant in its own right, merely an aspect of the greater cyclical nature of the world.

It sounds to me like your understanding of rebirth directly contradicts your understanding of Anatta. Have you considered your current understanding of rebirth may be inaccurate?

What is a better understanding @w33t? Perhaps you can suggest where the OP is going wrong or what the correct understand of rebirth is that is not in tension with anatta as you understand it?

@YesheTenley in my experience, death and rebirth is a never ending cycle that happens continuously in this eternal moment. What we call sights, sounds, tastes, feelings, smells, and thoughts are constantly appearing and disappearing. There is nothing that appears within awareness that is constant. Consistency is imagined through pattern recognition. For example when the sight of a couch appears, its imagined to be familiar, even though the present sight is unlike any that has ever appeared before. The senses and thought do not require conditions to appear. They do not require a self to appear.

Dharma is the truth, but must look into it logically, and understand fully, why waste time trying to learn advance Dharma when you cannot comprehend the basics properly? Best you learn about the basics.? You say you can accept 'Anatta' how can you accept 'Anatta' and why should you accept 'Anatta' ?
Then you say that you can rationally accept that suffering arises from attachment, how is that possible for you to accept that when you have not understood how 'Anicca' contributes suffering to arise?   Why do you say that meditation helps when you do not fully understand what it is for?
Then you talk about a migration of  consciousness, but do you understand anything at all what the consciousness is and why it arises? And you also mentioned transmigration - there is nothing migrate at all,  if you understood the meaning of 'Anicca' and 'Anatta' you will soon realise your questions are not valid. This is the very reason Buddha put 'IGNORANCE' on top of the chart of Dependant Origination.
May I suggest you to learn first the meanings of the first five precepts properly and see if you can live according to those, when you start to live that way I guarantee you, everything will fall into place and then you will understand what Dharma is trying to say,  Dharma is given to us by Buddha because he discovered that there is a never ending process of life existence and all life existences are un-satisfactory and brings suffering, therefore to stop the process of suffering in the cycle of existence (to liberate from suffering)  he found a method and that is why he is known as Buddha. He gave his discovery to the world free of charge.
Everything in the cosmos exist moment to moment and keep changing, everything has three phases,  nothing remains the same way,  they appear, exist, disappear , same way  Birth, decay, Death, and this process continues., new environment, new appearance new existence. (eg. A man lives happily, the next minute his whole life does a 180 deg change say he was charged with murder that he did not commit and end up in prison, his life situation has now completely changed, different environment, unknown people, fear, etc… although he is not clinically dead he is not the same person anymore because of his thinking pattern has changed. Re-Birth process is similar, because it is through your habits that give rise to cause (karma) and karma gives rise to a new existence, the new existence gives rise to more habits through new experiences,  and this cycle goes on and on, basically it is the information super highway what we call samsara is the base for all existences.
By the way Buddha never had a religion and he rejected all 'Ism's and Dharma was and still is and will always be a living system, extremely simple system but you need tremendous amount of effort to practice.
‘Kalama Sutta’
“Now, Kalamas, don’t go by reports, by legends, by traditions, by scripture, by logical conjecture, by inference, by analogies, by agreement through pondering views, by probability, or by the thought, ‘This contemplative is our teacher.’ When you know for yourselves that, ‘These qualities are skillful; these qualities are blameless; these qualities are praised by the wise; these qualities, when adopted & carried out, lead to welfare & to happiness’ — then you should enter & remain in them.
The quote comes from Kalama Sutra (AN 3.65) and is often taken out of context, hence misunderstood.
People of Kalama found themselves bombarded by tens of spiritual teachers, each claiming authority and expertise in spiritual matters. These teachers' doctrines were rather different from each other, but each was presented as The truth. Each teacher seemed quite certain of himself and was able to articulate the teaching logically.
When Buddha on one of his tours around the country arrived at Kalama and presented his teaching, the citizens honestly told him, that what he posits as The truth, to them looks like yet another teaching. "Is there any way" they asked, "that we can figure out which of these teachings is real?"
And that's when Buddha gave his famous answer, the point of which:
It is by its results that a teaching should be evaluated.
•   A teaching can be elaborate and logical, with precise definitions. According to some people's preconceptions, these are the marks of a true teaching.
•   A teaching could be profound, deep and mysterious. Some people assume, if chasm is deep and they can't see the bottom, there must be something in there.
•   A teaching could match student's view of the world, e.g. the scientific worldview, or a spiritual worldview, or both. Many people interpret Kalama Sutra this way, that they should not believe a teaching unless it matches "common sense". They don't seem to realize that what they assume as common sense is in fact the very tangle of preconceptions that holds them in Samsara.
•   A teaching could go against student's preconceptions and introduce a completely new theory of everything. Some students are very excited about such esoteric teachings and their eyes glaze over teachers that, in their opinion, profanate Dharma by assuming it speaks about our everyday lives.
•   A teacher can look confident and speak well, or be soft-spoken and funny, like Dalai Lama. Many people find it hard to relate to a teacher who mismatches their archetype of Sage or Wise Old Man.
According to Buddha, all these are secondary factors, that can't be used as identifying markers of Sat-Dharma (True/Eternal Law/Tradition). Instead, it is by the effects it brings out, both in student's psyche as well as in the world, that a teaching should be measured.
Sath-Dharma is famously good in the beginning, good in the middle and good in the end.
Good in the beginning means, even the outermost layer of Dharma, the one seen by others ( non-Buddhists), has good influence on people. Even the laypeople who don't really practice, but are merely guided by basic Dharma principles, benefit from it. They find that Dharma not only happens to match their highest secular moral and wisdom, but that while secular moral is often too flexible, the compass of Dharma never wavers. When followed at large, True Dharma must lead to reduced suffering and increased harmony in daily lives of common people.
Good in the middle means, when someone practices a slightly superficial version of Dharma, without fully understanding it yet, it greatly reduces amount of suffering one generates inside and around. Student learns to watch his mind and recognize its state, learns to stay mindful of the body and notice arising emotions, learns to not let harmful thoughts and emotions control him, learns to let go of attachments and preconceptions. This leads to increased quality of life, as the student can now stay cool through various life challenges.
Good in the end means, one eventually arrives at Liberating Realization, whereby one is no longer dominated by arbitrary formations, but can instead juggle formations freely.
While some kind of "Good in the end" is obviously the goal of all alternative teachings, it is "Good in the middle" and "Good in the beginning" that is a characteristic mark of True Dharma.
So when Gautama said, "do not believe anything I say until you can prove it by yourself" (or however you want to put it), this is what he meant. He did not mean we should reject a teaching unless it matches our preconceptions. He meant we should evaluate a teaching by its effect on our lives.
The proof is in the pudding.

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Incoherent Meditations
Seemingly, despite my questions, I still am encountering great difficulties. I've been meditating for perhaps 1-2 years in a haphazard fashion, on the metro, walking places, while doing things. Perhaps the word "meditation" is inappropriate, it was more like a mindfulness or perhaps a mixture of different types of reasoning, contemplation and attempts at realizing certain facts. Overall, I feel like I've accomplished absolutely nothing in these two years, and that I've wasted much time.
I would say the only thing I want now is to be able to stop all this, whether to meditate properly, or even to understand how to reverse the conditioning that I unwittingly imposed upon myself. I'm asking if anyone knows or conjectures a means by which I could achieve this, and perhaps some tips on what I should avoid doing to create this massive incoherence in my mind.

I am sorry but I don't understand what you're saying about reversing the conditioning. Just let your thoughts go. They are not important.

Don't just create a problem when there is none. This happened with me.

I also experienced similar situation where progress stalled through there was some initial progress where I tried out different traditions and techniques until I fell into the right track.
What could be happening in your practice is:

the wrong technique as meditation
a technique not suitable for you
the technique the wrong way

You have to figure out what went wrong.
A formal meditation supposed to strengthen your practice while engaging in daily activity. Mindfulness should be present throughout your daily activities. This is not what is going wrong.
Scrutinise the instructions and read to validate your understanding. In case you practice what is unsuitable for you a wrong technique. Best starting point is to look into the Suttas which highlights the practice which might work for the majority. You will need a bit of trial and error to figure out what works.
Basic outline of what needs to be done is the you should be aware of:

arising and passing
of sensation (pleasant, unpleasant, neutral) that is felt.

You can analyse this at the level of sensations from corporeal body as Mahabhuta.

vedana can certainly be connected with three of the four mahabhuta (elements), which are collectively known as rupa. These three are pathavi (earth element), tejo (fire element) and vayo (air element). Apo (water element) is too subtle to touch, or is so subtle that one finds difficulty in feeling it. Now, to illustrate the connection between vedana and the three bhuta, when you have a cramp, which is the tightening of the muscles, it is pathavi; when you feel temperature of the body it is tejo; when you suffer from pain in the stomach, it may be something to do with vayo, for when the air in your stomach is stuck and finds no vent to get out, you have that experience of unpleasant feeling called pain. The connection of vedana with these mahabhuta makes it all the more manifest and helps one realize its true nature.

Source: Vipassana and Vedana as Understood by a Novice by U Tin Lwin
Similarly any of the 4 Satipatthana. More details and references can be found here and here.
Also look at the links between the 4 Noble Truths and what is felt. For details see Vedana and the Four Noble Truths by VRI.
Also the link between Dependent Origination and what is felt. For details see: Vedana in Paticcasamuppada
Understand the relationship with what is felt (pleasant, unpleasant and neutral) with hindrances. See answer here for more details.

It sounds like you are experiencing runaway mindfulness.  At the very least, you're falling into discursive thought (as you say "reasoning" and attempts at "realizing certain facts").  Neither of those is really Buddhist in practice.  They're also not going to get you any closer to true realization.  I really think you need to slow things down.  A little samatha might go a long way in settling your mind and help keep it a bit more collected.  Once you've established some degree of calm, you might have more success with a mindfulness practice.
And like Suminda says, make sure you learn correct technique before you have at it again!

Anapanasati( mindfulness of breathing, breathe in, know it, breathe out, know it), sit in a quiet place and try to keep on any time you have. One may think it is the beginning of meditation(or just to concentrate, because it is so simple) but later surely will know: it is the very beginning, middle and the end i.e. you will see the Nibbana, no other way, the ONLY way and never missed.

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text:
Why Buddha said "if you want to remove small/light codes from Vinaya"?
In Maha-parinibbana Sutta, Buddha said "if Sanga want, after I completed Parinibbana, you can remove small and/or light codes from Vinaya". Why Buddha said this approval to Sanga? Is it a kind of testing to future members of Sanga whether they can completely comply the Vinaya or not? This approval may lead to breakage of Sanga to Mahayana, Theravada, and more. This approval, is it being served as a reference to other parties of Sanga who cannot comply the vinaya completely not to be blamed by normal people? Or is it just a question to just Sanga at the time of been said and this was the way to be. Or any purpose of this approval? Or is it totally not approval, and just question?

If the Buddha had totally approved the removal of certain rules then the Buddha would have explicitly mentioned which ones.
In the 1st Council the monks could not decide what extent was minor rules, so ultimately decided to keep all the rules with the fault of not clarifying being with Ven. Ananda when the Buddha mentioned this.

The original sangha had no rules. The rules arose in response to specific situations. The number of such rules is given as 150 in the suttas, but there are 227 rules for male monastics in the Theravadin Vinaya. The Buddha was dissatisfied with the state of the sangha toward the end of his life. He refused to appoint a leader of the sangha after his death. He also stated that the lesser and minor rules could be abolished after his death.
Contrary to many modern monastics, who boast concerning the number of rules they follow, Mahakassapa said that the number of rules is inversely proportional to the state of spirituality in the sangha. If you analyze the pratimoksha, you will see that there are really only about ten fundamental rules (see my "Synopsis of Vinaya"). These are not that different from the precepts of the vow of the bodhisattva. Elsewhere the Buddha makes it clear that following rules is a relatively minor thing, contrary to the aforesaid monastics. This provides context for the Buddha's injunction, which was not honoured by his immediate successors. The rules of the Vinaya became a major point of controversy that led to the first schism.

Alexander, I think that is somehow an oversimplification. The (historically) first rule can be seen as that one, when he told his five friends to call him no more "brother". Aside of this, the ascetic community must have had at least informal rules, and having studied with at least two gurus in their sanghas (before his awaking) he knew the basic rules for running a community - and even if we do not read about this in early suttas I think it is reasonable to assume such. The later rules, on top of them and (perhaps) formalizing, mentioned in the suttas are from the famous conflict situations

I'm not sure how my answer is an "oversimplication" based on your comment, but I don't disagree with anything in your comment.

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text:
How to cure "if I blamed someone who we should not blame"?
According to the Buddha's teaching, how can one cure the offense of "blaming someone who we should not blame"? Here "someone" means from Sotapanna to Arahant; and "blame" means to accuse that what he/she does (or what he/she does not do) is done (or not done) in not a good way.
Is there any chance that normal people like me can blame Sotapanna - Arahant in a way he/she is not doing good?
If I should not blame and I blamed, how can it be cured so that I do not accuse improperly and do not make any harm to myself?
What is Buddha's teaching said about it?
What is proper way of curing to completely remove the burden of accusing the noble being? How it can be done in details?
Even if it is not in direct Buddha's teachings or in Dhamma, is there any example or something that had been done before in some past experiences?

There's no completely escaping the consequences of a bad deed unless the Karma becomes defunct. But you can do much to mitigate the effects.

If the monk is alive, apologize in person. If he is dead, think of him and apologize.
Do good deeds. ex: offering alms to the monk you insulted or to the Sagha as a whole.
Do meditation. ex: Spreading Metta to all beings, practicing the four foundations of mindfulness etc.

All of the above will help a great deal, but only a Buddha would know what exactly you should do to make a Karma defunct, if it is possible.

In Buddhism, everything is forgivable.
If a noble person is practising their realisation, there are no grounds for us to blame them since the noble person will not be harming us.
If a noble person unintentionally harms us, they should ask for our forgiveness & we should forgive them.
If we think a person that intentionally harms us is a "noble person", we are mistaken because a noble person cannot intentionally harm others, even out of good-will (e.g., to kill a murderer) because the eightfold path prohibits all violence for any reason.
It is common for ordinary people to mistakenly believe practitioners of forms of black magic are noble people.
That is why if a monk kills any human being for any reason, including mercy killing, they are expelled from the monastic order.
AN 9.7 says:

...an arahant monk whose mental fermentations are ended, who has reached fulfillment, done the task, laid down the burden, attained the
  true goal, totally destroyed the fetter of becoming, and who is
  released through right gnosis, cannot possibly transgress these nine
  principles.
[1] It is impossible for a monk whose mental fermentations are ended to intentionally deprive a living being of life.
[2] It is impossible for a monk whose mental fermentations are ended to take, in the manner of stealing, what is not given.
[3] It is impossible for a monk whose mental fermentations are ended to engage in sexual intercourse.
[4] It is impossible for a monk whose mental fermentations are ended to tell a conscious lie.
[5] It is impossible for a monk whose mental fermentations are ended to consume stored-up sensual things as he did before, when he was a
  householder.
[6] It is impossible for a monk whose mental fermentations are ended to follow a bias based on desire.
[7] It is impossible for a monk whose mental fermentations are ended to follow a bias based on aversion.
[8] It is impossible for a monk whose mental fermentations are ended to follow a bias based on fear.
[9] It is impossible for a monk whose mental fermentations are ended to follow a bias based on delusion.

this is my own thinking and not from any sutta, why not become a noble follower yourself?  Fighting fire with fire.
Then Dhamma of the noble followers will come to you naturally.   Example, if someone causes you some displeasure, you have the knowledge that pleasure/displeasure is not caused by others but because of Phassa (contact), vedana arises.  This is not a common knowledge. all other beings in the universe who have not heard dharma think that pleasures or displeasure caused by others, by self, no cause...etc.
when one applies Pratītyasamutpāda to everyday life, there is really no one to blame.  Clinging to 5 skandha as self is Dukkha

It is the intention behind the criticism that will determine the weight of one's kamma. So check youself everytime, what you do, say, and think to the other person, whether they're for sincere positive, and constructive criticism for their own good, or it's simply for the sheer unwholesome pleasure of criticism. The safest approach, is to thoroughly investigate and gather all the info. before saying anything to anyone. Refer to MN 58 for some guideline on proper speech.


text:
Retaining courage during mediation
Everyone in this little community is awesome, and the people contributing have helped me in the path tremendously. Thank you.
My question is this, how can a person remain calm and courageous during sessions? When meditating, I've been reaching very intense states that become hard to handle. I've been getting the sensation that I am melting into this state of pure love. I get scared because it's just so unbelievably intense, and feel that the fear that arises is a hindrance. Have any of you experienced this? If so, what did you change that allowed you to handle it?

I found this talk by Leigh Brasington a day after asking this question. Link to video. He discusses this specific state around the 21 minute mark. Hope this can be helpful to someone in the future that stumbles upon this question.

The fear is instinctual (due to the ego fading/dissolving/melting somewhat). This instinctual fear can only be endured with composure & with the full acknowledgment it will pass. After the mind has experienced the impermanence of the fear a few times and is fully confident the fear will pass, the fear will no longer be a hindrance because it will be easily passed through.
If the mind thinks about/rationalizes the fear too much, the concentration will be lost. You have answered your own question, since the only real method is composure & courage.
From the Pali suttas:

What if I, in whatever state I'm in when fear & terror come to me, were to subdue that fear & terror in that very state?' So when fear &
  terror came to me while I was walking back & forth, I would not stand
  or sit or lie down. I would keep walking back & forth until I had
  subdued that fear & terror. When fear & terror came to me while I was
  standing, I would not walk or sit or lie down. I would keep standing
  until I had subdued that fear & terror. When fear & terror came to me
  while I was sitting, I would not lie down or stand up or walk. I would
  keep sitting until I had subdued that fear & terror. When fear &
  terror came to me while I was lying down, I would not sit up or stand
  or walk. I would keep lying down until I had subdued that fear &
  terror.
Bhaya-bherava Sutta: Fear & Terror

Thank you for the reply, the answer was very insightful. Just want to take a little time here to say I greatly respect you. Many hours have been spent just going through your answers to questions. You are spreading the dharma well my friend.

Because the "I" still interferes with those meditative states. Do not hold on to self-identity and your meditation would go much smoother:

Bhikkhus, how do you conceive it: is form permanent or impermanent?" — "Impermanent, venerable Sir." — "Now is what is impermanent painful or pleasant?" — "Painful, venerable Sir." — "Now is what is impermanent, what is painful since subject to change, fit to be regarded thus: 'This is mine, this is I, this is my self'"? — "No, venerable sir."
"Is feeling permanent or impermanent?...
"Is perception permanent or impermanent?...
"Are determinations permanent or impermanent?...
"Is consciousness permanent or impermanent?" — "Impermanent, venerable sir." — "Now is what is impermanent pleasant or painful?" — "Painful, venerable sir." — "Now is what is impermanent, what is painful since subject to change, fit to be regarded thus: 'This is mine, this is I, this is my self'"? — "No, venerable sir."
"So, bhikkhus any kind of form whatever, whether past, future or presently arisen, whether gross or subtle, whether in oneself or external, whether inferior or superior, whether far or near, must with right understanding how it is, be regarded thus: 'This is not mine, this is not I, this is not myself.'
"Any kind of feeling whatever...
"Any kind of perception whatever...
"Any kind of determination whatever...
"Any kind of consciousness whatever, whether past, future or presently arisen, whether gross or subtle, whether in oneself or external, whether inferior or superior, whether far or near must, with right understanding how it is, be regarded thus: 'This is not mine, this is not I, this is not my self.' ~~ SN 22.59 ~~

sounds like what Buddha described as Piti (litteraltraslation is "excitement").  It comes with first jhanna.  it will be gone when enter 3rd Jhanna.

"There is the case where a monk — quite withdrawn from sensuality,
  withdrawn from unskillful qualities — enters and remains in the first
  jhana: rapture and pleasure born from withdrawal, accompanied by
  directed thought and evaluation. He permeates and pervades, suffuses
  and fills this very body with the rapture and pleasure born from
  withdrawal. There is nothing of his entire body unpervaded by rapture
  and pleasure born from withdrawal.
"Just as if a skilled bathman or bathman's apprentice would pour bath
  powder into a brass basin and knead it together, sprinkling it again
  and again with water, so that his ball of bath powder — saturated,
  moisture-laden, permeated within and without — would nevertheless not
  drip; even so, the monk permeates, suffuses and fills this very body
  with the rapture and pleasure born of withdrawal. There is nothing of
  his entire body unpervaded by rapture and pleasure born from
  withdrawal...

Be very aware of the changing nature of the sensations. And stay equanimous. Awareness (Pali: sati) and equanimity (Pali: upekkha) are the two wings of vipassana. Awareness of the changing sensations and equanimity of the experience of the law of anicca create wisdom (Pali: panna).

Bhikkhu, not necessarily to be a monk, who is afraid of Samsara(round of birth and rebirth), must do insight meditation with relentless effort that even all your energy run out, (be courageous like a lion). If one feels scared, just meditate scare/fear , just keep faith in Dhamma, it will go away because nothing is permanent. If still there is, remember the Arahan(one of the nine Qualities of Buddha) , and take refugee in Buddha again and again. Sometimes one is in the sixth stage of  nana (knowledge of fear), then keep on meditating leads to higher stage of nana and overcome it.

" For every action , there is an equal and opposite reaction ". This is a Fundamental law of Physics and is applicable to Human Mental activities.  If fear comes as a reaction to the feeling of Intense Love ,you have to Absorb these reactions . Assume that these fears are Unreal and Temporary,wait for some time and allow the fears to die down .And then , continue with your daily activities--in this case--Meditation.


text:
Is it helpful to stay single if I am not going to become a monk?
In most Buddhist traditions, monks are expected to follow an austere life without being married or involved with another person, but what about lay people who are not going to become monks?
Personally, I've reached what most people consider mid-life and I'm getting too old for any monastery to accept me, but I'm sure this question applies to lay people both older and younger.
I am under the impression, monks are required to be single because of the attachment and desire that arises when being involved. However, a good number of monks do not live among lay people which I imagine would make it easier. It seems like lay people would benefit though, and when their time on Earth has expired also make it easier to let go.
However, I don't have a teacher nor do I have access to one so I wanted to reach out to the community to see what their impressions were. My guess is that it is helpful but a personal choice that differs from individual to individual. I mean 'helpful' in the sense of being closer to Nirvana or Enlightenment.

thanks for asking such a practical question. I am 29, now and in same feeling. I think the best you can ever get on this is already within you. waiting for your choice in terms of a perfact anwser.

Here are all the suttas which mention marriage  http://www.accesstoinsight.org/search_results.html?cx=015061908441090246348%3Adj4lxnh4dda&cof=FORID%3A9%3BNB%3A1&ie=UTF-8&q=marriage&sa=Search

It seems like Accesstoinsight.org carries Google ads now.

What matters is what you do when you are single. If being single means using prostitutes to satisfy one's sensual desires, it could lead one further away from the path as compared to a person who restrains himself to one woman and practices loyalty, kindness, doing good deeds together etc.
But if being single means being celibate and using one's free time for meditation etc., one would generally have a higher chance of progressing in the path as compared to the average family man.
In the time of the Buddha, there were monks who went to forest for seclusion and ended up having sex with monkeys. So what really matters is what you do when you are single.

Does using prostitutes necessitate not being loyal/kind/etc? Or is it just more likely that one who regularly visits them will not have these traits?

Usually people use prostitutes for sexual gratification & to get the most out of the money they spend. There's very little room for wholesome mindsets during such encounters

As you've said, it all varies from individual to individual. The bottom line is one should not do what other people expects one to do. Instead, do whatever is best for oneself in terms of peace, happiness, and most important, suitable and favorable conditions to the learning, contemplation, and practice of the Buddha Dhamma. Please see DN 29, where the Buddha taught about the 10 kinds of disciples that make up His spiritual community. Out of those 10, 4 are from lay folks like us: celibate laymen, householder laymen, celibate laywomen, and householder laywomen. Give it a careful read and it could be a useful guide to help with your inquiry. Good luck..

Maybe you can try to guard your sense doors. This can be the best help you can get.
Ayya Khema (Who Is My Self: A guide to Buddhist Meditation'):

"Here a monk, on seeing a visible object with the eye, does not grasp
  at its major signs or secondary characteristics. Because greed and
  sorrow, evil unskilled states, would overwhelm him if he dwelt leaving
  this eye-faculty unguarded, so he practices guarding it, he protects
  the eye faculty, develops restraints of the eye-faculty. . .  On
  hearing a sound with the ear, . . . on smelling an odor with the nose,
  . . .  on on tasting a flavor with the tongue, on feeling an object
  with the body, . . . on thinking a thought with the mind . . . He
  experiences within himself the blameless bliss that comes from
  maintaining this Ariyan guarding of the faculties. In this way . . . a
  monk is a guardian of the sense-doors.
When the eye sees, it simply registers color and shape. All the rest
  takes place in the mind. For instance, we see a piece of chocolate.
  The eye sees only the brown shape. It is the mind that says: "Ah,
  chocolate! That tastes delicious - I want a piece!" Not to grasp at
  the major signs or secondary characteristics is to stop the mind from
  doing exactly that.  We can practice this easily with anything we
  either very much like or very much dislike . . .  . . .   If we are
  easily swayed by what we see, the best thing to do is to recognize the
  sense-contact and stop the mind at the perception, the labeling. It is
  very hard to stop it before that. So, for example, if we see a person,
  or even think of a person, for whom we have hate or greed, someone we
  either dislike or long for intensely, we should practice stopping at
  the label, person friend, male, female. Nothing further. The rest is
  our desire. That is what is meant by guarding the sense-doors. Our
  senses are our survival system. It is much easier to survive if we can
  see and hear than if we are blind or deaf. Most people assume,
  however, that the senses are there in order to provide them with
  pleasure. We use them in that way and become angry when they fail to
  do so. We then blame the trigger. If someone displeases us, we blame
  that person. It has nothing to do with the other person, who, like us,
  is made up of the four elements, has the same senses, the same limbs,
  and is looking, as we are, for happiness. There is nothing in that
  person that is producing displeasure. It is all in our own mind.
  Exactly the same applies when we think another person will provide us
  with pleasure . . . There is no reason to look to that person for
  pleasure or blame then for not providing it. All we have to do is see
  "person". Nothing more. There are so many "persons" in this world, why
  should we allow this particular one to arouse our syndrome of
  desire-distaste? If we guard our senses, we guard our passions, which
  enables us to live with far greater equanimity. We are no longer on
  that endless seesaw; up, when we are getting what we want, down, when
  we are not, which induces a continual inner feeling of wanting
  something that just escapes us. Nothing that is to be had in the
  world, anywhere, under any circumstances, is capable of bringing
  fulfillment. All that the world can provide are sense-contacts -
  seeing, hearing, tasting, touching, smelling, and thinking. All are
  short-lived and have to be renewed, over and over again. This takes
  time and energy, and here again it is not the sense-contact itself
  that satisfies us. It is what the mind makes of it. Guarding the
  sense-doors is one of the most important things we can do, if we want
  to lead a peaceful, harmonious life, untroubled by wanting what we do
  not have, or not wanting what we do have. These are the only two
  causes of dukkha; there are no others. If we watch our sense-contacts
  and do not go past the labeling, we have a very good chance of being
  at ease."

Thank you - the passage indirectly answers the question but is helpful.

I respect the other answers given, however I would like to approach the answer from another angle.
I suggest not making a decision one way or the other. The Buddhist precepts ask you to refrain from irresponsible sexual actions; these can be understood as actions that  bring harm to some one—including yourself. Not seeing prostitutes would be just one aspect of this. Not cheating on someone or being with someone else’s partner are other obvious examples. But a relationship is much more than sex. And here is where my answer differs from to many of those above. Being in a caring responsible relationship allows you to practice loving kindness and compassion in many ways that are not available to someone who is not. If children are involved—either your own or from a partner’s life before you—then the opportunities are expanded even more. There are few roles more important then being a good compassionate parent preparing a child for this world, and for them to be good people in this changing world. Another is being a reliable partner for someone as well. But this doesn’t mean giving up the Dharma or converting those around you to it.
Personally in my early twenties I decided to experience voluntary celibacy for a period and I chose 6 months. I found it a very useful period. But I did not do it again, I’m now married with a stepson. Life provides many opportunities to practice the Dharma and improve myself as a person. I would argue more opportunities than if I was a celibate lay person.
I suppose it comes down to how you approach the Dharma—is it towards personal awakening or more towards becoming a Bodhisattva? Be aware of extremes and their attraction—remember the Buddha chose the middle path.


text:
Thousand fold world system
In the Kosala Sutta (AN 10.29) it quotes the following:

"As far, bhikkhus, as this thousandfold world system extends, Mahābrahmā there ranks as the foremost."

So basically as far as the sun and the mood illuminates the light a world system is there. 4 hells, human world , 6 humans and so forth. And thousand of this is called a thousandfold world system.
So according to the above quote, does this mean that there is only one Maha Brahma to all these thousand world systems?  Or are there 1000 Maha Brahmas in this thousand world system? Or is it just 1?

31 planes of existence coming together is one universe. In each universe there is a Maha Brahma in each Brahma plane.

... a thousand brahmā worlds. ...

According to the Abhidhamma there are infinite world systems. See: Abhidharmartha Pradipika By Amaradasa Rathnapala.
Also see: The Buddhist Concept of World by Bhikkhu T. Seelananda and also this answer.

Thank you for the info. Would you be kind enough to provide the source of the abhidhamma text that talks about infinite universes?

See: Abhidharmartha Pradipika By Amaradasa Rathnapala.

Also see this answer for more related information: http://buddhism.stackexchange.com/a/17978/295

It is said Loka is a World

“ As far, bhikkhus, as this thousandfold world system extends, Mahābrahmā there ranks as the foremost.
(3) “There comes a time, bhikkhus, when this world dissolves. https://suttacentral.net/an10.29/en/bodhi

Lokantagamanasuttam

Bhikkhus, I say that the end of the world cannot be known, seen, or reached by travelling. https://suttacentral.net/sn35.116/en/bodhi

Rohitassasuttam

"I tell you, friend, that it is not possible by traveling to know or see or reach a far end of the cosmos where one does not take birth, age, die, pass away, or reappear."
"It is amazing, lord, and awesome, how well that has been said by the Blessed One: 'I tell you, friend, that it is not possible by traveling to know or see or reach a far end of the cosmos where one does not take birth, age, die, pass away, or reappear.' Once I was a seer named Rohitassa, a student of Bhoja, a powerful sky-walker. My speed was as fast as that of a strong archer — well-trained, a practiced hand, a practiced sharp-shooter — shooting a light arrow across the shadow of a palm tree. My stride stretched as far as the east sea is from the west. To me, endowed with such speed, such a stride, there came the desire: 'I will go traveling to the end of the cosmos.' I — with a one-hundred year life, a one-hundred year span — spent one hundred years traveling — apart from the time spent on eating, drinking, chewing & tasting, urinating & defecating, and sleeping to fight off weariness — but without reaching the end of the cosmos I died along the way. So it is amazing, lord, and awesome, how well that has been said by the Blessed One: 'I tell you, friend, that it is not possible by traveling to know or see or reach a far end of the cosmos where one does not take birth, age, die, pass away, or reappear.'" https://www.accesstoinsight.org/tipitaka/an/an04/an04.045.than.html

Attend closely
and I will speak."
"I will, lord," replied the venerable Ānanda
to the Exalted One,
who said:

"As far as moon and sun move in their course
and light up all quarters with their radiance,
so far extends the thousandfold world-system.

Therein are a thousand moons,
a thousand suns,
a thousand Sinerus, lords of mountains:
a thousand Rose-Apple Lands,[4]
a thousand Western Ox-wainsa
thousand Northern Kurus,[5]
a thousand Eastern Videhas;[6]
four thousand mighty Oceans,
four thousand Mighty Ruler[7],
a thousand Four Great Rulers,[8]
a thousand heavens of the Thirty-Three,
a thousand Yama worlds,
a thousand heavens of the Devas of Delight,
a thousand heavens of the Devas that delight in creation,
the same of those Devas that delight in others' creations,
and a thousand Brahma worlds.
This, Ānanda,
is called:
'The system of the thousand lesser worlds.'
A system a thousandfold the size of this
is called:
'The Twice-a-thousand Middling Thousandfold World-system.'
A system a thousandfold the size of this is called:
'The Thrice-a-thousand Mighty Thousandfold World-system.'
Now, Ānanda, if he wished it,
the Tathāgata could make his voice heard
throughout this last-named world-system,
or even further,
if he chose."

"Pray, lord, how could that be done?

"In this connexion, Ānanda,
the Tathāgata suffuses with [208] radiance the Thrice-a-thousand Mighty Thousandfold World-system.
When its inhabitants perceive this,
then the Tathāgata would give utterance
and make the sound heard.
That is how he would do it." http://obo.genaud.net/dhamma-vinaya/pts/an/03_threes/an03.080.wood.pts.htm

‘It’s impossible for two perfected ones, fully awakened Buddhas to arise in the world system [lokadhātuyā] at the same time.

There is this passage in

That in the world by which one is a perceiver of the world, a conceiver of the world—this is called the world in the Noble One’s Discipline. And what, friends, is that in the world by which one is a perceiver of the world, a conceiver of the world? The eye is that in the world by which one is a perceiver of the world, a conceiver of the world . The ear … The nose … The tongue … The body … The mind is that in the world by which one is a perceiver of the world, a conceiver of the world. That in the world by which one is a perceiver of the world, a conceiver of the world—this is called the world in the Noble One’s Discipline.

Note that here the formulae is that there is a world and in that world (the conceived) there are worlds (doors; that though which one perceives & conceives).
There are other texts maybe related to this part

“Bhikkhus, as far as sun and moon revolve and light up the quarters with their brightness, so far the thousandfold world system extends.

Darkness Sutta

Mendicants, there is lokantarikam (interworldly?) Darkness, so utterly dark that even the light of the moon and the sun, so mighty and powerful, makes no impression / doesn't reach.” https://suttacentral.net/sn56.46/en/sujato

I'll just leave it here but the expression that there is no electromagnetic radiation (light) inbetween the worlds, where worlds are subjective frames of reference is basically a key tenent of special relativity and would seemingly be agreeable by sutta method.
I also think that the natural reading interpretation is that Mahabrahma is the foremost among Brahmas, like the foremost of men would be included among men, so i think Mahabrahma is one of the Brahma.


text:
In Buddhism, does "God" exist or not?
In Buddhism, is the existence of "God" accepted, or not? If not, what is the reason?

Within the Tripitaka, the oldest and most authentic memories of the Buddha's words, the Essential God, or simply Essence, is neither affirmed nor denied. This is explained in The All Embracing Net.
http://www.accesstoinsight.org/tipitaka/dn/dn.01.0.bodh.html
Metaphysical positions can never be more than speculation. The Essential Form that God needs to either Be or Not-Be is simply not evident within our empirical domain, nor within the domain of reason and logic. Mathematics even forbids Essential Truth within all formal systems:
https://en.m.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
If you read Nagarjuna's Fundamental Wisdom of the Middle Way, the deeply logical discussion of the Buddha's message, Essence within our empirical experience is denied evidence.
The Buddha was careful not to speculate beyond where perception and evidence were possible. Because if God Is (and all the mess that option 2 entails for him in Plato's Parmenides), and He Is beyond reach of our empirical experience, Independent of all of this, then we are like the man looking for the most beautiful woman in the land:
http://www.sacred-texts.com/bud/dob/dob-13tx.htm
(this is not an affirmation of Not-Being, Nihilism of any sort)
The theoretical insight of the Buddha is now merging with scientific evidence with regards to the absence of Essence within experience. Background independence, conservation laws, flatness of local space time, even relational quantum mechanics, all point to a much more profound understanding of all this:
https://suttacentral.net/en/mn1
If you are curious how modern philosophy approaches the Buddha's message, http://plato.stanford.edu/entries/buddha/#NonSel is interesting!

It depends on what you mean when you say "God" and "exists". In Buddhism all entities (objects) are abstractions or models, conceptual labels on top of mind-made constructs. So all things really exist in one sense, but don't really exist in another sense. God is like that too. If by God we mean "The Absolute" then it in one sense it obviously exists (since everything is included in it) but then in another sense The Absolute is just an abstraction so it doesn't really exist. It certainly exists as concept or model in our mental world.

On point answer!

In Buddhism, is the existence of "God" accepted, or not? If not, what
  is the reason?

"God" has many meanings depended on what religious or cultural context this word referred to. Yet to make it simple, God, as a being who is in a higher ranking than man, does exist, in Buddhist teaching. The Christian God, and Brahma, both exist, so are many of those mentioned in different cultures.
In Buddhist teaching, there are many worlds and many forms of beings, which mainly categorized into 6 groups: 1. Celestial beings (they are gods; beings who accumulated great merits almost subdued greed, anger, ignorance, arrogance, distrust but still controlled by desire) 2. Asuras (some may call them devils, demons; the beings who accumulated great merits but are deeply grabbed by anger and arrogance) 3. Men 4. Animals 5. (hungry) Ghosts 6. Hell beings.  The Celestial beings (sometimes Asuras) many cultures referred to as gods. They have more abilities due to their subduing of contamination (the 5 main pollutants that hindered one's abilities: greed, anger, ignorance, arrogance and distrust). Thus they can exhibit supernatural powers such as being able to alter weather, turn stone into gold, etc. And of course they have clarity of vision they can see anything from any distance and be in any place in a blink of seconds, etc. And their body material is a different made from man, we cannot see them, and depends. Some of the gods such as Indra and Brahma (one is called "Conch-tufted Brahma" because of His hair style??) are mentioned in some of the sutras, they were audiences in the Buddha's sermons or pledged to uphold the teachings. Often said that Indra is the God in Christian faith however this is just one of the sayings.


text:
During meditation how do I focus
What is the best way to focus?
And also how can I stop passing judgment on myself during? Its really difficult.

What is the best way to focus?

Initially bring your mind to the object. Now note (without any verbalization or visualisation) any sensations anything you feel about the object. This can be a something like touch of air in breathing or more particularly related to the elements or positive, neutral and or negative perceptions and sensory reaction if mental. (additional information see: Pahāna Sutta, Dhātu Vibhaṅga Sutta, Titth’ayatana Sutta) Now knowing the mind is on the object try to bring the mind to the object again, i.e., retain the mind on the object.

And also how can I stop passing judgment on myself during? Its really difficult.

Judgement about self to arise you need through about self. Just focus on the object so all metal activity is directed at being focused until such time you mind just sticks with the chosen object and thoughts do not arise. This way you can aviod judgements about self.

To focus, you remember to keep the mind on the meditation object (breathing) &, when the mind strays, just bring it back to the object.
When the mind strays, it is not "you" that is intentionally straying but the mind itself therefore it is not logical to judge "yourself" since the mind is the problem rather than "yourself".
If the mind is straying a lot and you are judging yourself a lot, you may have to train the mind new ways of acknowledging the straying, such as mentally noting: "The mind is straying"; "the mind has craving"; "the mind has anger"; "the mind is restless"; etc or simply: "straying"; "craving"; "anger"; "restlessness", etc.
In this way, instead of judging "yourself" & attributing blame upon "yourself" unnecessarily & unwarrantedly, the mind is judged or the states of mind are judged.
There are preliminary meditation techniques for beginners such as the popular Mahasi Sayadaw method that use this  kind of mental noting. You may check out these methods or, otherwise, read about how the Buddha dealt with his straying mind in the sutta on Two Kinds of Thinking.

One of the biggest obstacles that beginners often run into is trying to focus.  This strategy is very artificial and is often extremely counter productive.  Seng Ts'an, the third patriarch of Zen states this plainly when he says "when you try to stop activity to achieve passivity, your very effort fills you with activity".  Don't do that.
Let's look at your mental ecology for a minute.  First, you are exerting effort and trying to achieve mental focus.  I don't know about you, but whenever I try to do anything, my body clenches, my breath gets shorter, and my mind resembles something like television static.  That's no way to enter samadhi.  To add to your problems, the amount of self-judging going on is just going to make matters worse.  Frankly, if you find yourself getting into that kind of try-critique-frustration loop, my best advice to you is to get up off the cushion, go for a walk, or make yourself a cup of tea.  Either of those are going to benefit you far more than staying on the cushion and beating yourself up.
You need to relax, my friend.  Meditation isn't like lifting weights or running track.  Crushing down on your meditation object is the exact opposite of what you want to be doing.  One-pointedness is something that arises all of its own.  It's not something that you create through effort.  All you can do is set the table.  Samadhi will show up to eat if he likes what you laid out for him.
I could give you some hokey metaphor - something about how the dishes on the table are moral discipline, the mashed potatos your karmic roots, or something equally dreadful.  I'm not because I think you've got enough swimming around in your head already.  Instead, my advice to you is to do nothing.  I'm serious.  Sit on the cushion or even a chair.  Plop down on a bed.  It doesn't matter.  Just spend thirty minutes there being completely still and doing absolutely nothing.  Don't count your breath.  Don't watch your thoughts.  Don't do anything!  Just hang out and don't move a muscle.  Easy peasy, right?  I'll bet you still struggle with it, but so long as you are doing nothing, you are doing it right.  If you find this boring, great.  In fact, I hope you do.  Boredom is what it feels like when your small mind is struggling to stay afloat in the ocean of Big Mind.  You want to drown that bastard.
This should be a regular form of self care.  We all spend way too much time doing.  Instead, make a point of don'ting.  Go for a hike.  Practice zhan zhaung (which is sort of what I'm recommending above).  Whatever.  It doesn't matter.  Just stop trying so damn hard.  Once you've don'ted consistently for a while, you should begin to have a better idea of what it is to passively watch or be with the breath versus attacking the poor thing.  And remember, the Buddha's path is good in the beginning, good in the middle, and good in the end.  It should never be a source of self loathing.  Just relax.

the best object for beginner to concentrate on is breathing, the reason is because you have it, you dont have to visualisation
when you meditate, keep focusing on your breath, dont control it, just let it breath by itself, when your mind start to wander around(not focus) then take a deep breath and start it all over again
good luck, you can do it!!!

There is so many hindrance to focus in  meditation. To overcome this, one must be truly modest first. To take refugee in three Gems and also to the meditation master. To ask for pardon for all the misdeed physically, verbally or mentally either knowingly or unknowingly. Also to forgive everybody for their misdeed to you. All the preparation made with strong belief, it is easier to get concentration. After that with the most comfortable, posture, start noting mindfully "breathing in, know it, breathing out , know it" focusing at the tip of the nose and no where else. Still difficult to focus, just remember Arahan or any qualities of Buddha which you prefer and after that resume meditation. It will surely help you.

Buddha talked about focusing on body (breath is part of the body) more often, than also he talked about 3 other foundations (Satipatthana)
you may focus your attention on any of the 4 foundation of mindfulness

body (breathing)
feeling (feeling of equanimity from meditation)
Consciousness (the awareness itself)
and dhammas (such rise and fall/unreliable/subject to destruction of each of the above)

If it is difficult to focus, please check

Did you take (at least) eight(8) precepts before meditation?
You have to please yourself that your day is good and you didn't break any of eight precepts so that any immediate judging yourself will not occur.
Are you doing meditation in quiet place where no sound distraction, no view distraction (the best practice is closing your eyes during meditation but any movement of people passing by, etc)?
Dedication to meditation (you have strong determination to commit meditation no matter what dangers and hinderances you need to overcome).

For the people with difficult to focus, it is good to start watching breathing-in and breathing-out. Most important fact is you need to breath normally and naturally. If you try to breath faster, you will get tired very soon. If you try to breath slower, you will get distracted.
The very first beginner can start counting breath, breath in and breath out count as one, another pair two and so on. Counting should not less than 5 and not more than 10 so that any mathematical distraction is not involved during meditation.
The next stage is just watching on breathing in and breathing out without counting. There might be some distractions happen in your mind and lose attention on breath-in and out. Then you need to know there is distraction. As soon as you know there is distraction the train of thoughts(or some other form of distraction) will cease. Then you need to put your attention back to breath-in and out again. You need to do this till you are master to putting your attention back as fast as you start distracted. Soon you can get full attention on Anapanasati.
Mind is very fast to tend to distraction even if you can do full attention to your breath-in and out. Then you need to do next level of Anapana. Get your attention to beginning of breath-in, middle (the air passing in your nose) of breath-in, end of breath-in; the beginning of breath-out, middle (the air passing out your nose) of breath-out, end of breath-out. If you can master this level of attention on Anapana, you can probably be free from focus problem in meditation.
For the next levels of meditation, you need to approach a good teacher who can closely guide you. People with different hobbies, perspectives, intentions and goals needs to be taken care of by individually. You definitely need to go meditation centers and need to do proper retreat. There you can nurture/practice your mindfulness, concentration, and whatever you are looking for.


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Purifying my mind
How can I purify my mind so that I can gain confidence and believe that I can study and practice efficiently? The major hindrance for that, what I believe, is that my subconscious mind is strongly saying that I am incapable.

It is just simple. The hierarchy is sila(at least five precepts for lay man),Samadhi(concentration),and insight knowledge. To achieve this in order, the very first thing is one's strong believe in Dhamma and the meditation master who can teach you in every detail. Just like inertia, if one not yet start , it is difficult to do so. But one of the encouraging fact is the will to do so if one has, it the very beginning. Throughout Sansara, all these defilement clinging to one mind but insight meditation if properly done no matter one has all the misdeeds except the great five ones. So as a very step go to a retreat you rely on where you can find a god master, and start this hierarchy relentlessly at least for 2 weeks. If you have the basic after that you can proceed at home whenever you find time. As mentioned earlier, just to start. If concentration moment accumulate from second to minute to hour , you will find the progress.

This is one of the Hindrances which is Doubt. The best way to overcome this is wise attention, i.e., see through vipallasa by means of Vipassana as mentioned in the other answer. [Nīvarana,pahana Vagga, Yoniso Manasikāra Sampadā Sutta, Vipallasa Sutta].
Also see: Nīvaraṇa by Piya Tan, The 5 Mental Hindrances by Piya Tan,

meditate, if you meditate correctly, you can gain relax mind that will help you to see the problem that you have right now and think of the solution, but focus on meditating not the problem when you meditate, for beginner the best object for beginner to concentrate on is breathing, the reason is because you have it, you dont have to visualisation when you meditate, keep focusing on your breath, dont control it, just let it breath by itself, when your mind start to wander around(not focus) then take a deep breath and start it all over again.
but if your real problem is your mindset that think negative about yourself, i offer you to read book:
1. The Magic of Thinking Big by David J.Schwartz
2. The Power of Habit by Charles Duhigg

I forget what the Nichiren practice is called ... Gosho? Daimoku? where you memorize the Korean Sanskrit of an entire surya (either the Expedient Means one or the Life-Span of the Thus Come One one or something) and recite it (usually in unison with the rest of the Nichiren Buddhist congregation) ... https://youtu.be/650WAvcZokA


text:
After Enlightenment - 7 Weeks (Sath Sathiya)
As we all have heard the Buddha spent 7 weeks after his enlightenment in different places and engaging in different activities.
But in the Mahavagga Pali (the first half) in Vinaya Pitakaya states that the Buddha only spent 4 weeks after enlightenment. And after that week only the Sahampati Brhama invited him to teach Dhamma to the world. (In the 7 weeks tradition this happened in the 7th week)
1st Week  - Under the Bodhi tree
2nd Week  - Under the Ajhapal Nuga(Goatherds’Banyan tree) tree
3rd Week  - Under the Midhella tree (The Mucalinda story)
4th Week  - Under the Kiripalu tree (The Tapssu Balluka story)
Also in the Sutra Pitakaya I found the following sutras as well.
These sutras talks about the first week.
Ud1.1
Ud1.2
Ud1.3
This sutra talks about the second week.
Ud1.4
This sutra talks about the third week
Ud2.1
4th week is anyways stated in the Mahavagga Pali.
But in the Pujawaliya written by Ven. Mayurapada Parivenadhipathi Buddhaputhra suggests that the Buddha spent 7 weeks after the enlightnment in different places and engaging in different activities. Sath Sathiya - 7 Weeks
If Vinaya Pitakaya is the one that is to be accepted (Vinaya and Dhamma are the Teachers after me - Buddha) then did the Pujawaliya writer exaggerate the 4 weeks into 7 weeks? Even in that book i didn't find any references to the Tripitakaya (Pali Canon) as well.

Very timely with all the interest around Pujawaliya in the radio and media.

There was a possibility that Buddha stayed in Uruvela forest more than four(4) weeks. It was mentioned in Mahavagga Pali, section 6, end of paragraph 10 but it did not specify exactly 3 weeks more or places of additional weeks. The last line of that paragraph here

did mention Buddha stayed more than four(4) weeks probably made the best timing for five foremost monks for first discourse.
If found about this more details in any sutta, I'll update here.

The additional information might be in the commentaries?

But him staying 4 weeks there and taking 4 weeks more to travel to Isipathana by foot seems legit. There's around 200-300 KM on the way. So taking a month to go there is possible.

Mahavagga Paliya, Kahnadaka clearly says Buddha stayed 5 weeks (not 4) after enlightenment.

First week - Under the Bo-tree. Templating on Patichchasamuppada. (Chain of causation)
Second week - Under Ajapal banyan tree. Meeting a brahamin
Third week - Under the cover of Muchalinda, the serpent.
Forth week - Under the Rajayatana tree. Meeting two merchants
Fifth week - Again under Ajapal banyan tree. Meeting Sahampathi brahma


text:
"The Blessed One", is it bad translation?
I found "The Blessed One" in English translation of Suttas. Because in Buddhism blessing is not legit/proper word to use to anyone, especially Buddha need not to be blessed, is it loosely translated or in purpose? I didn't see any part of Sutta that one can be blessed or a meaning that blessing can be done so far. What should be proper translation of "The Blessed One" in Sutta if it is bad translation?

A couple of dictionary translations:

Bhagavā: venerable, fortunate, blessed, sublime. Generally designates the Buddha.

Bhagavant (adj. n.) [cp. Vedic bhagavant, fr. bhaga] fortunate, illustrious, sublime, as Ep. and title "Lord." Thus applied to the Buddha (amhākaŋ Bh.) and his predecessors. Occurs with extreme frequency; of fanciful exegetic explns of the term & its meaning we mention e. g. those at Nd1 142=Nd2 466; Vism 210 sq.; DA i.33 sq. Usual trs. Blessed One, Exalted One.

Also, perhaps as an etymology:

Bhaga [Vedic bhaga, bhaj, see bhagavant etc.] luck, lot, fortune, only in cpd. dub˚ (adj.) unhappy, unpleasant, uncomfortable It 90; DA i.96 (˚karaṇa). -- bhaga (in verse "bhagehi ca vibhattavā" in exegesis of word "Bhagava") at DA i.34 read bhava, as read at id. p. Vism 210.

Here are some definitions of "blessed" from an American dictionary (perhaps Protestant culture):

Simple Definition of blessed
  : having a sacred nature : connected with God
  : very welcome, pleasant, or appreciated
Full Definition of blessed

a :  held in reverence :  venerated
  b :  honored in worship :  hallowed
  c :  beatific
:  of or enjoying happiness; specifically :  enjoying the bliss of heaven —used as a title for a beatified person
:  bringing pleasure, contentment, or good fortune

I think you might see that many of these definitions are suitable.
You're right that it's probably not blessed in a Catholic sense, i.e. sanctified via the agency of a priest.

The word is 'bhagava', from 'bhaga'.
You may prefer 'Fortunate One' if you believe 'blessings' can only be given by a person or a god.

Yeah; in the original text of The Bible (when Jesus is giving The Beatitudes, saying "Blessed are the 'this' & Blessed are the 'that"), the word they translated "Blessed" actually means something more like 'deliriously, abundantly joyfully-happy.'

Well the term is translated as The World Honored One in Chinese

Wikipedia says, The tenth epithet is sometimes listed as "The World Honored Enlightened One" (Skt. Buddha-Lokanatha) or "The Blessed Enlightened One" (Skt. Buddha-Bhagavan). "World Honored" is presumably Lokanatha rather than Bhagavan; I'm don't know why they're listed together. The root Loka means "world" or something like that.

The Chinese name for "Buddha" is ”世尊", which means “The Respected One”。 “One” is with relation to the "法界"， or "The Dharma Realm" in English or scientifically speaking， “The Universe”


text:
How did Buddha deal with people that were sexually attracted to him?
Is there any known case, where someone was sexually attracted to Buddha?

http://www.buddhanet.net/e-learning/buddhism/lifebuddha/2_9lbud.htm

I recall there is a story (which is probably not from the Pali suttas but from a commentary) where a woman was in love with the Buddha so the Buddha used psychic power to make his appearance age before the woman, similar to the commentary story below:

Janapadakalyāṇī became a nun because her eldest brother (the Buddha) had renounced his kingdom to become a monk, his son Rāhula had become
a monk, her own husband, Nanda, had become a monk, and her mother, had
all gone forth. Due to her great beauty, she became known as Rūpanandā
Therī. Hearing that the Buddha deprecated beauty, she avoided visiting
him. However, hearing the lavish praise of the Buddha by the nuns and
women devotees, she wished to see and hear the Buddha, so one day went
with the other nuns, sitting behind them. The Buddha used his psychic
powers to create a vision of a beautiful young woman fanning him.
Rūpanandā was captivated by her beauty. The Buddha then caused the
young woman to age gradually, slowly taking on the form of an old
woman, then one dying in agony. Finally he made her body became a
bloated corpse, pecked at by crows. Becoming thoroughly disgusted,
Rūpanandā reflected that her own body was the same. Then the Buddha
uttered the above verse and Rūpanandā gained Arahantship.
http://www.aimwell.org/11-jara.html

There was a monk named Vakkali who was attracted to buddha's physical appearance. He became a monk only to be able to stay closer to buddha and gaze at his beauty. Buddha warned him firmly by saying the following,

Vakkali, what is there in looking upon this filthy body? Vakkali, he who sees the Dharma sees
  me

and sent him away from his presence. Vakkali was depressed by this and wanted to commit suicide, however at the moment of committing suicide buddha was able to help him get enlightened.
Source:http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/8.8-Vakkali-S-s22.87-piya1.pdf