text:
What are the views considered "false convictions", and why are they false?
According to Brahmajāla Sutta, there are "62 False Convictions" (Views) regarding the Past & Future.

False Convictions Regarding the Past

Eternity of the Self and World,
Partial Eternity and Partial Non-Eternity of the Self and World.
Finitude and/or Infinitude of the World.
Evasive Strategies Resorted to by "Eel-Wrigglers" [or Endless Equivocators].
Chance Origination of the World.

False Convictions Regarding the Future

Conscious Post-Mortem Survival.
Unconscious Post-Mortem Survival.
Neither Conscious Nor Unconscious Post-Mortem Survival.
Nihilistic Views Regarding Post-Mortem Survival.
Nirvana Here and Now.

Why are they considered as false?

The following quotes are extracts from the Pali Deeganikaya  Brahmajāla Sutta: The All-embracing Net of Views.

Speculations about the Past (Pubbantakappika)
"There are, bhikkhus, other dhammas, deep, difficult to see, difficult to understand, peaceful and sublime, beyond the sphere of reasoning, subtle, comprehensible only to the wise, which the Tathāgata, having realized for himself with direct knowledge, propounds to others; and it is concerning these that those who would rightly praise the Tathāgata in accordance with reality would speak. And what are these dhammas?
"There are, bhikkhus, some recluses and brahmins who are speculators about the past, who hold settled views about the past, and who on eighteen grounds assert various conceptual theorems referring to the past. And owing to what, with reference to what, do these honorable recluses and brahmins frame their speculations?

Eternalism (Sassatavāda): Views 1–4
Partial-Eternalism (Ekaccasassatavāda): Views 5–8
Doctrines of the Finitude and Infinity of the World (Antānantavāda): Views 9–12
Doctrines of Endless Equivocation (Amarāvikkhepavāda): Views 13–16

"In the fourth case, owing to what, with reference to what, are some honorable recluses and brahmins endless equivocators, resorting to evasive statements and to endless equivocation?
"Herein, bhikkhus, a certain recluse or a brahmin is dull and stupid. Due to his dullness and stupidity, when he is questioned about this or that point, he resorts to evasive statements and to endless equivocation: 'If you ask me whether there is a world beyond — if I thought there is another world, I would declare that there is. But I do not take it thus, nor do I take it in that way, nor do I take it in some other way. I do not say that it is not, nor do I say that is neither this nor that.'
"Similarly, when asked any of the following questions, he resorts to the same evasive statements and to endless equivocation:
A.
  •   2. Is there no world beyond?
  •   3. Is it that there both is and is not a world beyond?
  •   4. Is it that there neither is nor is not a world beyond?
  B.
  •   1. Are there beings spontaneously reborn?
  •   2. Are there no beings spontaneously reborn?
  •   3. Is it that there both are and are not beings spontaneously reborn?
  •   4. Is it that there neither are nor are not beings spontaneously reborn?
  C.
  •   1. Is there fruit and result of good and bad action?
  •   2. Is there no fruit and result of good and bad action?
  •   3. Is it that there both is and is not fruit and result of good and bad action?
  •   4. Is it that there neither is nor is not fruit and result of good and bad action?
  D.
  •   1.Does the Tathāgata exist after death?
  •   2.Does the Tathāgata not exist after death?
  •   3.Does the Tathāgata both exist and not exist after death?
  •   4.Does the Tathāgata neither exist nor not exist after death?
"This bhikkhus, is the fourth case.
"It is on these four grounds, bhikkhus, that those recluses and brahmins who are endless equivocators resort to evasive statements and to endless equivocation when questioned about this or that point. Whatever recluses or brahmins there may be who resort to evasive statements and to endless equivocation, all of them do so on these four grounds or on a certain one of them. Outside of these there is none.
"This, bhikkhus, the Tathāgata understands... and it is concerning these that those who would rightly praise the Tathāgata in accordance with reality would speak.

Doctrines of Fortuitous Origination (Adhiccasamuppannavāda): Views 17–18

Speculations about the Future (Aparantakappika)
"There are, bhikkhus, some recluses and brahmins who are speculators about the future, who hold settled views about the future, and who on forty-four grounds assert various conceptual theorems referring to the future. And owing to what, with reference to what, do these honorable recluses and brahmins frame their speculations?

Doctrines of Percipient Immortality (Saññīvāda): Views 19–34

"There are, bhikkhus, some recluses and brahmins who maintain a doctrine of percipient immortality[10] and who on sixteen grounds proclaim the self to survive percipient after death. And owing to what, with reference to what, do these honorable recluses and brahmins proclaim their views?
"They proclaim: 'The self is immutable after death, percipient, and:
A.
  •   1. material
  •   2. immaterial
  •   3. both material and immaterial
  •   4. neither material nor immaterial
  B.
  •   1. finite
  •   2. infinite
  •   3. both finite and infinite
  •   4. neither finite nor infinite
  C.
  •   1. of uniform perception
  •   2. of diversified perception
  •   3. of limited perception
  •   4. of boundless perception
  D.
  •   1. exclusively happy
  •   2. exclusively miserable
  •   3. both happy and miserable
  •   4. neither happy nor miserable.'
"It is on these sixteen grounds, bhikkhus, that those recluses and brahmins who maintain a doctrine of percipient immortality proclaim the self to survive percipient after death. Whatever recluses or brahmins maintain a doctrine of percipient immortality, all of them do so on these sixteen grounds or on a certain one of them. Outside of these there is none.
  "This, bhikkhus, the Tathāgata understand... and it is concerning these that those who would rightly praise the Tathāgata in accordance with reality would speak.

Doctrines of Non-percipient Immortality (Asaññīvāda): Views 35–42
Doctrines of Neither Percipient Nor Non-Percipient Immortality (N'evasaññī-nāsaññīvāda): Views 43–50
Annihilationism (Ucchedavāda): Views 51–57
Doctrines of Nibbāna Here and Now (Diṭṭhadhammanibbānavāda): Views 58–62

This teaching on the 62 (false) convictions is from the Brahmajala Sutta (in the Pali Digha Nikaya -- adapted herein from Walshe).

At the end of this Sutta the Buddha declares that this teaching may be known by the following names:
• The Net of Advantage (Attha = Artha)
  •   The Net of Dhamma
  •   The Supreme Net
  •   The Net of Views
  •   The Incomparable Victory in Battle
In his glossary entry on the “sixty-two convictions” in the Vimalakirti Sutra, Thurman states:

These ... consist of all views other than the "right view" of selflessness. All sixty-two fall into either one of the two categories known as the "two extremisms": "eternalism" (sashvatavada) and "nihilism" (ucchedavada).

[However,] for Vimalakirti’s take on this subject, see his exchange with Manjushri below.
Vimalakirti’s Take on the 62 Convictions (From p. 44 of Thurman’s translation):
Mañjushri: Householder, where should emptiness be sought?
Vimalakirti: Mañjushri, emptiness should be sought among the sixty-two
  convictions.
Mañjushri: Where should the sixty-two convictions be sought?
  Vimalakirti: They should be sought in the liberation of the
  Tathagatas.    Mañjushri: Where should the liberation of the
  Tathagatas be sought?    Vimalakirti: It should be sought in the prime
  mental activity of all living beings.
Mañjushri, you ask me why I am without servants, but all Maras and
  opponents are my servants. Why? The Maras advocate this life of birth
  and death, and the bodhisattva does not avoid life. The heterodox
  opponents advocate convictions, and the bodhisattva is not troubled by
  convictions. Therefore, all Maras and opponents are my servants.

Buddha's conclusion (From Wikipedia)

"The Tathagata knows these sixty-two views. He also knows the dhamma
  which surpasses them. Knowing that dhamma, he does not view it in the
  wrong way. Since he does not view it in the wrong way, he realizes by
  himself the extinction of defilements (i.e., greed, anger, and
  ignorance of the Four Ariya Truths).
Buddha finally concludes the exposition of these 'wrong' beliefs by
  stating that these (62) beliefs, if they are believed, will certainly
  cause agitations and cravings. It implies that the beliefs come to
  conclusion due to the inability to see the truth, as they are seized
  by craving (clinging), agitated by longing (feeling).
The Buddha further explains that the beliefs are originated from
  Contact (Phassa) as the cause. The contact is a phenomenon when the
  perception recognised an object beyond our Self. Then, from this brief
  (like lightning in the sky, Nagasena analogued in Milinda Panha)
  event, rise up feelings.
Buddha states that there are no possibilities of feeling without
  contact. Thus, according to the law of Twelve Related Chain of Cause
  and Effects (Pratitya-samutpada), the people who believe in one of
  many of these sixty-two beliefs, will end up in round cycle of
  sufferings; as they have not found the truth on the cease of
  sufferings.
Due to their faith, they will experience feelings as a result of
  repeated contact through the six sense bases. In them feeling gives
  rise to craving; craving gives rise to clinging; clinging gives rise
  to current existence (upapatti bhava) and the kammic causal process
  (kamma bhava); the kammic causal process gives rise to rebirth; and
  rebirth gives rise to ageing, death, grief, lamentation, pain,
  distress and despair.
The Buddha states that Monks who have realized and understood the
  origin of contact of the six senses, and escaped the round of
  sufferings, would see Dhamma (Truth) of Precepts (Śīla), Concentration
  (Samadhi) and Wisdom (Pańńa) which surpassed all the wrong beliefs.
The Buddha then makes an analogy of a fisherman using a fine-meshed
  net to catch the fish in the pond. The fish represent the ascetics who
  cling to their beliefs. They will rise and sink in the pond, but in
  the end will unavoidably be caught in the net. Whereas the Buddha, who
  stand outside the net has found the truth and has transcended the
  cycle of suffering.
The Brahmajala Sutta ends with this quotation:

"When the Bhagava had delivered this discourse, the Venerable Ananda
    addressed him thus: "Marvellous indeed, Venerable Sir! Extraordinary
    indeed, Venerable Sir! What is the name of this exposition of the
    dhamma?"
"Ananda!" said the Bhagava, "Bear in mind that this exposition of the
    dhamma is called Atthajala, the Net of Essence, as well as Dhammajala,
    the Net of the Dhamma, as well as Brahmajala, the Net of Perfect
    Wisdom, as well as Ditthijala, the Net of Views, as well as
    Anuttarasangama Vijaya, the Incomparable Victory in Battle."

Thus said
  the Bhagava.

For full detail analysis refer Bikku Bodi's book.

This 'Brhmajala Sutta"(The Supreme Net) is the first in thipitaka. It is grate to know Jesus was  preach about this net of fishing to his first disciples.

Shrawaka what do you mean when you said “Jesus was preach about this net of fishing to his first disciples” I don’t think Jesus preach the Brhmajala sutta to his disciple because this is a Buddhist sutta


text:
What does the word "world" mean in the Buddhist context?
The Hemawatha Sutta Is a (very old)? sutta in "Sutha Nipatha" of KN, Just after the first sermon of "Dammachakka". 'Hemawata' (General of yakka's ) is also a listener of that. He give the news about Buddha to his friend 'sathagira' and both came to meet load. That Dialog is in this sutta. some extract of it as follows

...
The Great One like a lion who lives alone,
among all pleasures he’s expectation-free,
let us draw near that we may ask of him
how to escape from the snarefulness of death.
O proclaimer of the Dharma, expounding it too,
one who’s beyond all dharmas’ Further Shore,
all fear and hatred you’ve utterly overcome
both of us then of Gotama inquire—
Hemavata
What co-arises with the world?
With what’s it make acquaintance?
The world grasps after what indeed?
Why’s the world afflicted?
Buddha
Six with the world do co-arise,
with six becomes acquainted,
the world’s attached to six indeed,
so, world’s by six afflicted.

The word 'world' is used in this verses and throughout the thipitaka.
What is the meaning of this word in this context?

In the context of presentations of dependent-arising as exposed in the Pali canon (as in, for instance, the quotes you give, or also Thanissaro Bhikkhu phrasing "There is nothing further for this world" at the end of the Adittapariyaya Sutta) I would trust Geshe Sonam Rinchen's explanation (in "How Karma Works"):

"The world," used to translate the Tibetan term jigten (jig rten), here has a very specific meaning because it refers to the five
  aggregates - forms, feelings, discriminations, compositional factors,
  and the six kinds of consciousnesses - which constitute our own and
  others' bodies and minds. The person[...] depends on these five
  aggregates, and it is in relation to them that we come into existence
  and disintegrate. Thus "basis of disintegration" is one meaning for
  jigten."

Hi Dorje Tenzin and welcome to Buddhism SE. We have a Guide and a Resource section for new users, that you might find useful.

'Namasthe'(Hello Banthe) Dorje welcome to SE. I am Terawadian from Sri lanka.

In the quote you have provided, the “world” is the six senses. The Buddha often used the word “world” in a metaphorical sense. Here are some Suttas where "world" is used in a similar way:
SN 35.82:

Then a certain monk went to the Blessed One and, on arrival, having bowed down to him, sat to one side. As he was sitting there, he said to the Blessed One: "'The world, the world' it is said. In what respect does the word 'world' apply?
  "Insofar as it disintegrates, monk, it is called the 'world.' Now what disintegrates? The eye disintegrates. Forms disintegrate. Consciousness at the eye disintegrates. Contact at the eye disintegrates. And whatever there is that arises in dependence on contact at the eye — experienced as pleasure, pain or neither-pleasure-nor-pain — that too disintegrates.
(repeat for ear, nose, tongue, body and mind-sense)

SN 12.44:

Dwelling at Savatthi. There the Blessed One addressed the monks: "I will teach you the origination of the world & the ending of the world. Listen & pay close attention. I will speak."
"As you say, lord," the monks responded to the Blessed One.
The Blessed One said: "And what is the origination of the world? Dependent on the eye & forms there arises eye-consciousness. The meeting of the three is 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, & despair come into play. This is the origination of the world.
(repeat for ear, nose, tongue, body and mind-sense)

I believe by "world" Buddha usually means the human civilisation, the society, the world of people:

This world is blind! There are so few who see things as they truly are.

...

Whose Dhamma is well-taught? Who has practiced well in this world? Who in the world is well-gone?"

...

He has made known — having realized it through direct knowledge — this world with its devas, maras, & brahmas, its generations with their contemplatives & brahmans, their rulers & common people.

...

He who in this world takes nothing that is not given to him, be it long or short, small or big, good or bad — him do I call a holy man.

...

This world completely lacks essence \ It trembles in all directions \ I longed to find myself a place \ Unscathed — but I could not see it.

...

He who having been heedless is heedless no more, illuminates this world like the moon freed from clouds.

...

Birth is ended, the holy life fulfilled, the task done. There is nothing further for the sake of this world.

I add my answer here.

Load Buddha has realize new visions. to communicate those ideas he need new words (for new concepts). Instead of defining new words he sometimes use them  with new meaning.(This is one example).
The word "world" has two usages, one for normal meaning as shown in Andrei's answer.
The other meaning is for new understanding about the 'world' we create by our self. It create by 'Six sense doors'.

Instead of using the word "world", '5 Aggregate', '12 Ayathana'(faculty),' 18 Dathu' or 'Dukka'(suffering), 'Bimba'(Illusion), 'Maya', etc also used to as appropriate.
In 'Hemawatha sutta' the dialog goes as..
....
Hemavata
The grasping—what is it then
by which the world’s afflicted?
When asked about this, please do speak:
how to be free from dukkha?
Buddha
The sensual pleasures five are taught
in the world with mind as six,
having let go of all desire for those,
be thus from dukkha free.
This for the world’s the leading out,
its “as-it-is” declared to you,
and this to you I do declare:
be thus from dukkha free.
...

source: Stack Exchange

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/

sort: votes


text:
Stream winner reborn as human
If someone enters the stream and becomes a sotapanna (stream winner), then he would have at most, seven lifetimes left. What the stream is, is defined in this answer as:

This noble eightfold path — right view, right resolve, right speech,
  right action, right livelihood, right effort, right mindfulness, right
  concentration — is the stream. (SN 55.5)

Thanissaro Bhikku further comments here that:

The coming-together of these factors is called the stream because it
  leads inevitably to two things, just as the current of a tributary
  will lead inevitably to a major river and then to the sea. In the
  immediate present, the stream leads directly to the arising of the
  Dhamma eye, the vision that actually constitutes this first awakening.
  Over time, the stream ensures that — in no more than seven lifetimes —
  one will be totally unbound.

The Dhamma eye is explained as the ability to see firsthand and experientially, the impermanence of the five aggregates and dependent origination.
My questions are (for rebirths as a human):

How does a reborn stream-winner know that he or she is one?
Does being a stream-winner in a previous lifetime, mean that one would be reborn into a devout Buddhist family in this lifetime or would definitely be led to the Dhamma eventually?
Is the reborn stream-winner's Dhamma eye open immediately at birth, or only after he or she encounters the Dhamma and returns back to the Noble Eightfold Path?
How is it guaranteed that the reborn stream-winner would not choose another path or another teacher, and drop out of the stream, considering that we do not remember previous lifetimes?
Please provide any info from the canon regarding this.

I could be wrong but a cula-sottapanna (truth seeker) is more likely to be born as human. A Sottapathi on death is less likely to choose human birth from the available choices (merely because for someone with pleasure seeking not fully extinguished the higher realms will be more tempting though ultimately unsatisfying). Reincarnate Bodhisattvas and Buddhas are another matter, their choice is deliberate and well informed. To question #3 - that description of dharma eye opening upon encounter with dharma sounds more like cula-sottapanna.

Do you have more info on different types of sotapanna?

Cula means lesser or little, it's a stage just before sottapanna. Search this forum for the term cula-sottapanna. I'm on my mobile, but I recall a question on this distinction was asked earlier.

http://buddhism.stackexchange.com/q/10019/715

How does a reborn stream-winner know that he or she is one?

Like a math or music prodigy who can easily recall their skillset through just some basic exposure to music or math, a stream-winner would be like that, if not much quicker or even instantaneous since Path/Fruit are supra-mundane skills, unlike mundane fields like math/music.
There're concrete metrics to determine a stream-winner. Details are available in many suttas like AN 9.12, AN 3.87, SN 12.41, etc. (ref: suttacentral.net)

Does being a stream-winner in a previous lifetime, mean that one would be reborn into a devout Buddhist family in this lifetime or would definitely be led to the Dhamma eventually?

S/he's to be born into any place(devas or humans) and time where the conditions are conducive to guarantee enlightenment at most 7 more life times.

Is the reborn stream-winner's Dhamma eye open immediately at birth, or only after he or she encounters the Dhamma and returns back to the Noble Eightfold Path?

The DhammaEye opens just once and will stay there when the noble disciple makes the breakthrough to StreamEntry, as said in AN 3.94:

"Bhikkhus, just as, in the autumn, when the sky is clear and cloudless, the sun, ascending in the sky, dispels all darkness from space as it shines and beams and radiates, so too, when the dust-free, stainless Dhamma-eye arises in the noble disciple, then, together with the arising of vision, the noble disciple abandons three fetters: personal-existence view, doubt, and wrong grasp of behavior and observances"

.

How is it guaranteed that the reborn stream-winner would not choose another path or another teacher, and drop out of the stream, considering that we do not remember previous lifetimes?

S/he won't switch path since they possess the Seven Great Reviewing Knowledges of Stream Entry (MN 48), one of which is: "Knowledge that no other recluse or brahmin outside the Buddha's Dispensation possessed of a view that one possess." Also see the child prodigy analogy mention in question 1.

Please provide any info from the canon regarding this

All refs. should be available on accesstoinsight.org or suttacentral.net

"The DhammaEye opens just once and will stay there when the noble disciple makes the breakthrough to StreamEntry, as said in AN 3.94." Therefore, babies are born as stream-enterers? Is there any evidence for this?

Here's my take on Qn 1, 3 and 4.

How does a reborn stream-winner know that he or she is one?

I don't think he or she will know unless they have attained the super-mundane powers like the recollection of past lives.

Is the reborn stream-winner's Dhamma eye open immediately at birth, or only after he or she encounters the Dhamma and returns back to the Noble Eightfold Path?

I think the Dhamma eye refers metaphorically to "seeing things as they truly are" i.e. the transient nature of things that are conditioned and conversely, a glimpse of the unconditioned, deathless state. I believe it is possible to see the deathless state again since they had already done so in their past lives, in fact, it maybe easier subsequently.

How is it guaranteed that the reborn stream-winner would not choose another path or another teacher, and drop out of the stream, considering that we do not remember previous lifetimes?

As santa100 had mentioned, knowledge can be either intellectual or intuitive. Once, the stream-winner sees the deathless state, it becomes an intuitive knowledge. They may not remember how they "saw the deathless state" in their past life but they will find reality to be very dis-satisfactory and struggle to seek the truth and a way out. So unless another path or another teaching contains the noble eightfold path that guarantees a way to attain Nibbana if not they would keep searching.
Although, it was never stated clearly in the suttas but I strongly suspect our Buddha before his enlightenment is a once-returner.

Sorry but rebirth doesn't exist. The universe is like a river and you can't step on same river twice. Everything keep changing. Even the human form didn't exist 200k years before. Anyone who believes nibbana is some kind of annihilation forever is deluded. And to think the universe exists just for humans to achieve nibbana is also deluded.

If the universe is a river and you get tired of it, ever wonder what's the shore like? Some wants to exist forever and fear annihilation, others want annihilation and hate existence. Both views, the Buddha says, leads to suffering. The deathless state never existed for anyone just as the shore didn’t exist to save any drowning person, it’s just there. Ehipassiko, the Buddha said; why not take up the challenge and see.

How does a reborn stream-winner know that he or she is one?

I think there is no sign or feeling about this. (Is there any sign or feeling when one reach the sotapanna even in this life time?)
The person "who attains a clear understanding of Three Marks of Existence and Four Noble Truths - and therefore cutting all doubts" as discussed in How does stream entry occur? becomes a stream enterer/Sotapanna.
They no longer have the first 3 fetters:

Belief in a self,
Doubts about the Buddha, his teachings or the
    noble sangha
Attachments to rites and rituals.
1). He does not have a view of "Atman" or permenent self (Identity
view). But he leaves the feeling of "I", My,Myself like "Avijja" or
ignorance.
2). If he finds (True) Buddha, Dhamma, or sangha he insistently
accepts them or understands what they realize.
3). He has the thinking power to select or understand what is write
or wrong to follow.

So Question 2 is not necessary.
For Question 3 & 4, The Dhamma eye (Understanding - clarity of thinking) is already with him and it is not lost (it improves continuously until final liberation).
Shall we consider this quart from this answer.

I believe there are probably a few ways to enter the stream.
I have entered the stream (as a lay person) Having no parents has
  really helped me to connect more deeply with the true nature of the
  way things are. I have had little conditioning, no preconceptions. As
  an autodidact, things have been revealed to me intuitively over many
  years.
For the last 20 years I have been going on walks (walking meditation)
  in the local forests asking questions. Nature has been my mother and
  father, my teacher. My eye has always been open on these walks. I
  realized that whatever arises would cease and it was heartwarming.
For me, being a stream winner is like this: When you have to overcome
  extreme trauma and appalling loss and you don't become cruel and you
  can forgive and you can let it go, you stop the cycle. You have done
  what you came here for and you transform. That and maybe 20 years of
  walking in woods...lol. I doubt I could do another 20! Be Well.

Sotapanna handbook
I have posted this book couple of times.  The publisher compiled nearly 50 suttas about stream enterer mentioned by Buddha. I'm wondering the same thing about reborn-stream enterer but couldnt find one mentioned by Buddha.  IMO, it is like parents sending a child who is on a right track off to college.  A child might get into some minor problems along the way but eventually will graduate. Parents dont have real concern knowing their child will reach his/her destination.  Bhuddha mentioned a few reborn places of Stream enterers and once returners but not more than one life so we dont know what will become of them next. (Ex. anathapindika, lady visaka, lady Sumana (once returner, anathapindika's daughter.  All reborn in heavenly realms)

I'm wondering the same thing about reborn-stream enterer but couldnt find one mentioned by Buddha

Ruben2020, and who ever might be interested in answers of you question:
Begin a time where the matter of "what makes one a stream-winner" seems to interest many and maybe because many "Noble-Born" are currently remind, meting they reminder, on their old path they have already gone, it's maybe worthy to recollect the teachings informations about this matter in a good manner and everybody should be encouraged here to point out possible mistakes and missing points.
❋ - Namo tassa bhagavato arahato sammā-sambuddhassa - ❋

➨ 1. How does a reborn stream-winner know that he or she is one?

Acutally in the frame set of the questions it would be unlikely the first, since why would one raise such a question if maybe not even aware of such as "steam-winner.
How ever, lets look what are the atributes that a Streamwinner "possesses". Of what [i]we[/i] know about what the Buddha said, being answered the question, "How does one know if having won the Stream or not?", the properly most similar question and most direct answer is found in the Maha-parinibbana Sutta, in an conversation mit Ven. Ananda Thera, the Bhikkhus have been used to ask for the attainments of others the Lord Buddha an he giving them a mirror to use to look for oneself:

The Mirror of the Dhamma
Ananda, I will give you the teaching called the Mirror of the Dhamma, possessing which the noble disciple, should he so desire, can declare of himself: 'There is no more rebirth for me in hell, nor as an animal or ghost, nor in any realm of woe. A stream-enterer am I, safe from falling into the states of misery, assured am I and bound for Enlightenment.'"
"And what, Ananda, is that teaching called the Mirror of Dhamma, possessing which the noble disciple may thus declare of himself?
"In this case, Ananda, the noble disciple possesses unwavering faith in the Buddha thus: 'The Blessed One is an Arahant, the Fully Enlightened One, perfect in knowledge and conduct, the Happy One, the knower of the world, the paramount trainer of beings, the teacher of gods and men, the Enlightened One, the Blessed One.'
"He possesses unwavering faith in the Dhamma thus: 'Well propounded by the Blessed One is the Dhamma, evident, timeless, [18] inviting investigation, leading to emancipation, to be comprehended by the wise, each for himself.'
"He possesses unwavering faith in the Blessed One's Order of Disciples thus: 'Well faring is the Blessed One's Order of Disciples, righteously, wisely, and dutifully: that is to say, the four pairs of men, the eight classes of persons. The Blessed One's Order of Disciples is worthy of honor, of hospitality, of offerings, of veneration — the supreme field for meritorious deeds in the world.'
"And he possesses virtues that are dear to the Noble Ones, complete and perfect, spotless and pure, which are liberating, praised by the wise, uninfluenced (by worldly concerns), and favorable to concentration of mind.
"This, Ananda, is the teaching called the Mirror of the Dhamma, whereby the noble disciple may thus know of himself: 'There is no more rebirth for me in hell, nor as an animal or ghost, nor in any realm of woe. A stream-enterer am I, safe from falling into the states of misery, assured am I and bound for Enlightenment.'"

There are other Suttas as well, which describe the attributs of a Streamwinner like in the Samyutta Nikaya, the SN 22.109: Sotaapanno Sutta — The Sotaapanna (' Stream-winner') or the Okkanta-samyutta — Entering

In this samyutta the Buddha explains the kinds of conviction and understanding that are required for the attainment of stream-entry. These short suttas share an identical structure, with each one focusing on a different aspect of experience (including the six senses, the six elements (dhatu), and the five aggregates). See also the Study Guides on stream-entry.

For example:

Cakkhu Sutta: The Eye
At Savatthi. "Monks, the eye is inconstant, changeable, alterable. The ear... The nose... The tongue... The body... The mind is inconstant, changeable, alterable.
"One who has conviction & belief that these phenomena are this way is called a faith-follower: one who has entered the orderliness of rightness, entered the plane of people of integrity, transcended the plane of the run-of-the-mill. He is incapable of doing any deed by which he might be reborn in hell, in the animal womb, or in the realm of hungry shades. He is incapable of passing away until he has realized the fruit of stream-entry.
"One who, after pondering with a modicum of discernment, has accepted that these phenomena are this way is called a Dhamma-follower: one who has entered the orderliness of rightness, entered the plane of people of integrity, transcended the plane of the run-of-the-mill. He is incapable of doing any deed by which he might be reborn in hell, in the animal womb, or in the realm of hungry shades. He is incapable of passing away until he has realized the fruit of stream-entry.
"One who knows and sees that these phenomena are this way is called a stream-enterer, steadfast, never again destined for states of woe, headed for self-awakening."

Itks maybe worthy to keep in mind that one needs to be very honest and proof very in detail all corners and dismantel any kind of quickly made judgement in its seriousity. That, how ever, requires to know, heaving heard the good teachings and understand clearly their meaning.
In regard of the fetters, a stream-winner has abounded the first three lower fetters of the ten fetters:

Self-identity views (sakkāya-diṭṭhi), identifying either the clinging-aggragats, or some, or one, or the sense-groups and their objects, some, or one, as being an lasting personality

Uncertainty or doubt (vicikiccha), meaning that one does no more seek for other teachers and teaching than the Buddha and his Dhamma, transported by the Savaka-Sangha.

Grasping at (wrong) precepts & practices (sīlabbata-parāmāsa), which are not conductive for the path and naturally follow conductive ones, relaying merely on the judgement of the own mind in regard of what is skillful and unskilful and no more need of authority be given precepts & practices in the judgements sphere of speech (verbalication) and bodily acts. With other words, deeds (in mind, speech and body) are now "controlled" in an inwardly manner and no more just and outwardly evaluation, in a ways of outwardly comparing appearances but by relaying on mindstates, e.g intention (cetana), not to mix with purpose) for it. While the wordling act merely fixed on the gain of certain fruit (vipāka), the Noble Ones view has turned to the quality of the action (kamma) when deciding what and how to do.

In regard of the virue of a streamwinner there are many views and disputes of which, how and if a streamwinner can break the basic four precepts and missconuct in consuming means which cause heedlessness.
Since the virtue of a Noble person of first degree is based on gained right view, he/she is not able to act strong unskillful deliberated. It might be that faulty acts happen in "affect", but even done he/she does not hidde such a fault and has a natural desire to confess it immediately.
How ever, he/she is in any case incapable to performing acts out of niyata micca ditthi, evil wrong view, and therefore incapable:

Quote from: Commentar to Sn 2.1

matricide,
2 patricide,
the murder of arahants (the Consummate Ones),
the shedding of the Buddha's blood,
causing schism in the Sangha, and
6.pernicious false beliefs (niyata micca ditthi).

In reagard of more detail of the virtue, my person gave recently some words on it in a discussion: Can there be Non-Attachment without Virtue?
In regard of recognition by others (and in the case one seeks for prove by others), is needed to know, that only a "person on integrity (e.g. a Noble One)" can know of anothers persons integrity (MN 110). The Sutta Paṭisalla Sutta: Seclusion contains also explainings here that:

It's through living together that a person's virtue... It's through trading with a person that his purity... It's through adversity that a person's endurance... It's through discussion that a person's discernment may be known, and then only after a long period, not a short period; by one who is attentive, not by one who is inattentive; by one who is discerning, not by one who is not discerning.

My person will end here in regard of the first question.

➨2. Does being a stream-winner in a previous lifetime, mean that one would be reborn into a devout Buddhist family in this lifetime or would definitely be led to the Dhamma eventually?

No, there are certain kinds mentioned and about the "leding", he she would definitly. On how: see answer question 3.
A steam-winner will be born

mostly 7 more times in the realm of the Devas (sensual world is no hight path is gained) or human world, here:
reborn in a good family
one-seeder
See AN 9.12 (Bhikkhu Bodhi):

... With the utter destruction of three fetters, this person is a one-seed attainer who, after being reborn once more as a human being, makes an end of suffering.

... With the utter destruction of three fetters, this person is a family-to-family attainer who, after roaming and wandering on among good families two or three times, makes an end of suffering.
...With the utter destruction of three fetters, this person is a seven times-at-most attainer who, after roaming and wandering on among devas and humans seven times at most, makes an end of suffering. ...

➨3. Is the reborn stream-winner's Dhamma eye open immediately at birth, or only after he or she encounters the Dhamma and returns back to the Noble Eightfold Path?

He/She maybe not conscious about having the " Dhamma-eye but like answered to question 4. is not in a strong danger to follow wrong teacher, in an amout that is not in the frame of the conduct and way of thinking of a Noble One.
Becoming aware about the Dhamma is also answered under question 4, which can have four general typs of reminder:

➨4. How is it guaranteed that the reborn stream-winner would not choose another path or another teacher, and drop out of the stream, considering that we do not remember previous lifetimes?

AN 4.191 indicates that one (here a previous monk appearing under Devas), if not reminding him-/herselves ("dies with a muddled mind") either is remembered by

comming in contact by him-/herself
a monk with special power
a Deva
a "familary" spirit

One is incapable to "fall out" or "follow other Paths", since - like in the 1. question answered - the fetter "doubt" is abounded. Now it might be possible that he/she trains within another religion als long and in so far, the views and practices contain the factors of the noble eightfold path. It's so possible, that if one maintains a "mystic" approach in regard of "God", it could be that one practices as Christian, or better under Christians, till one meets the good Dhamma, for example, or in certain other sects, but will maybe oppose certain things there naturally.
So it is guaranteed by or through his/her previous deeds, the fullfilement of bringing together all pathfactors and the hearing of the good Dhamma with proper attention in advanced.

➨5. Please provide any info from the canon regarding this.

The answers should already countain the sources. One of the best collection of Suttas might be the Studyguiden Into the Stream - A Study Guide on the First Stage of Awakening, by Bhante Thanissaro, which will provide many additional informations and teachings, not only for gifted "Noble-Born's" but also for Dhamma-, Faith- and most suggested for those "with little faith" for now, to not only gain more of it, but also put all means into practice and finish a task so less are capable, bringing behind the most of all burdens to be for ever behind.

Association with people of integrity is a factor for stream-entry.
Listening to the true Dhamma is a factor for stream-entry.
Appropriate attention is a factor for stream-entry.
Practice in accordance with the Dhamma is a factor for stream-entry.
— SN 55.5

May all beings have share of the merits here done with the support and sacrifies of many and may the Devas inform those not having seen or heard yet.
Anumodana!
You may find a possible extended or modified answer here, being also invited to comment, discuss or what ever insired: On Being of Noble birth - Stream winner reborn as human
(Note: This is a gift of Dhamma, came into being by goodness received and gratitude and so given by goodness and gratitude, and not meant for commercial purpose and trade in the world, or any wordily gain.)

this sutta explains the mechanism of how one makes progress in dhamma after he/she is reborn in heaven.  AN4.191 (i couldn't find English translation, but you could change language from pali-english text to Russian and then use google translation to English).   BAsically the sutta teaches us, if one remembers Buddha's teaching (even just one verse) and accepts it, memorizes it by heart. Even if that person passes away with cloudy mind (without sati), he or she will be born in deva world and will remember Buddha's teaching in 4 different ways.  He  will make progress (becomes stream enterer) in that realm.

English translation https://suttacentral.net/an4.191/en/sujato?layout=plain&reference=none&notes=asterisk&highlight=false&script=latin

If someone enters the stream and becomes a sotapanna (stream winner), then he would have at most, seven lifetimes left

For me, this is a very questionable interpretation of the teachings.
Suttas such as SN 13.1 say a stream-enterer has around 1/100,000th of the ordinary suffering of an ordinary person.
It seems ridiculous that a stream-enterer could extinguished 99,999/100,000 of suffering in one lifetime but required six more lifetimes (at most) to extinguish the other 1/100,000th of suffering.

Is the reborn stream-winner's Dhamma eye open immediately at birth, or only after he or she encounters the Dhamma and returns back to the
  Noble Eightfold Path? How is it guaranteed that the reborn
  stream-winner would not choose another path or another teacher, and
  drop out of the stream, considering that we do not remember previous
  lifetimes?

Very logical & intelligent questions here, which support the case the 'seven-life-times' idea is a misinterpretation of what the Buddha taught.
In many sutta, such as SN 56.47 & AN 4.123, the 'human state' is equated with understanding the four noble truths & the deva realms equated with reaching the meditative jhanas.
When the suttas state the stream-enterer has more time as a 'human' & 'god' ('deva'), it may refer to what is stated in SN 56.47 & AN 4.123.
It seems ridiculous that a 'noble person', that is considered enlightened, would wander around for another 560 years (7 x 80 year life) to attain full-enlightenment.

The suttas seem quiet regarding questions 1 and 2, other than to say that a sotapanna will have no more than seven rebirths in the human and deva realms. Also, I have never heard or read of any teacher who contradicted the suttas in this regard.
According to what I've read, beings in the lowest deva realm have lifespans lasting millions of years. (Google: 31 planes of existence dhammawiki)
Once the dhamma eye is open, it never closes. This prevents falling away. The power of a sotapanna's citta, or one could say, the power of the sotapanna's realization, overcomes all obstacles that would lead to leaving the Eightfold Path. A sotapanna arises when the kilesas of the first three fetters are uprooted and destroyed. They will never arise again. The sotapanna's mind and the Dhamma become inseparable, or one could say, the same, tho in a limited sense at this stage.
Ajahn Pasanno and Ajahn Amaro cover this in detail and with plentiful canonical citations in their free book available online: The Island. Available here:  http://www.amaravati.org/dhamma-books/the-island/
In addition, you can hear a lecture on the issue here: https://www.youtube.com/watch?v=AGz34CJQCfk
Your questions are reasonable, but don't spend too much time in speculation. It disturbs samatha and sati. And keep in mind the Kalama Sutta when you read replies. People tend to provide their own favorite ideas-- who among them have their Dhamma Eye open?
forgive any typos

It would be good if you could post the parargraphs from Ajahns Pasanno & Amaro you hold to be relevant. This is a question & answer forum. Thanks. With metta

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/

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text:
In the five paramitas, what is patience and why is it on the list?
I understand the English dictionary definition of patience, the ability to wait.
When reading about the paramitas, it seems like this word in the original language covered forbearance, i.e. the ability to suffer other people's anger and insults calmly, in addition to the sense of be capable of waiting.
Was patience added to the list because of the huge time required to achieve full Buddhahood? Or is patience a sloppy translation and is this paramita really about forbearance?
Anyhow, in personal practice, if it means the ability to wait, then it isn't actionable since, patient or not, there is little risk of me reaching full Buddhahood on any given day.

Patience is actually a great translation.  Think of the etymology of the English word:

patience (n.) - c. 1200, "quality of being willing to bear
  adversities, calm endurance of misfortune, suffering, etc.," from Old
  French pacience "patience; sufferance, permission" (12c.) and directly
  from Latin patientia "patience, endurance, submission"

Patience is to suffer the slings and arrows of outrageous fortune.  It is to endure, without complaint.  When one is patient, they turn headlong into discomfort and do so with a mind marked by equanimity.  Think of the countless lives of the Buddha - all of the suffering he had to endure in order to achieve nirvana.  His was a patience par excellence.
Patience is an indispensable virtue on the road to Buddhahood and absolutely deserves to be on this list!

Hmm, dictionary.com says you're right. My definition of patience is #2, the ability to put up with suffering is #1. Oddly, I can't remember anyone using it that way.

I have friends who are Catholic priests.  I hear it used that way all the time.  Truth be told, I think it's pretty instructive.  It's one of those little tidbits from comparative religion that is actually useful!

Patience is a mind that abides in its natural state unaffected by harm and suffering. It strongly abides in the Dharma.
The division is threefold:

Patience bearing suffering
Patience not paying heed to those who cause harm
Patience stable in the Dharma

To be a fully qualified perfection, it must be conjoined with the wisdom realizing emptiness and effortless bodhicitta. The perfection of patience is so perfected on the level of the 3rd ground (bhumi) on the supra-mundane path.
The six perfections are related to the two accumulations of (1) merits and (2) wisdom that lead to buddhahood. The accumulation of merits is related to the first three perfections: generosity, ethics and patience. (that partly explains why patience is a perfection)
Indeed, from the point of view of the result: the fruit of patience [and so forth] is buddhahood. The two essential causes for that fruit are: (1) myself practicing patience, and (2) the enemy that is the object of patience. One can not achieve enlightenment in isolation, since it depends on others. As in Shantideva:

6.108 Because I am able to practice this, He is worthy of being the very first to be given The fruit of my patience, For in this way he is
  the cause of it.

The relationship between ethics (2nd perfection) and patience (3rd perfection) is such that if one wants to practice the three aspects of ethics, it will involve hardships. This is where patience comes into the picture. We need to learn how to endure those hardships. (that also explains that it is a perfection, since it is in that sequence)
In terms of benefits, as anger is greatly disadvantageous since it destroys roots of virtue (the meaning of which is discussed at length by Tsongkhapa in the Middle Length Lam Rim and the Chen Mo as well), patience, on the other hand, is the direct antidote to anger. Thus, it prevents one from getting his roots of virtue destroyed: As in Gyaltsab Je's commentary to Shantideva:

Although one meditates the virtue of generosity and the like for eons,
  they are destroyed by     the fire tongue of anger. Therefore, one needs
  to generate the force of patience again and   again and not give anger
  any chance.

Patience has other benefits as well, such as:

Being a way to purify negativities
Being “the best way to accumulate merit” (I quote here one of my teachers, Kyabje Geshe Gyaltsen)
Increasing our qualities and happiness (i.e. if one did not react with anger but practiced patience the mind would not be affected by the harm and pain)
Being a way to take the essence of this precious human rebirth

That cultivating patience is a way to accumulate merit is shown in Shantideva:

6.103 If by my own fault I am not patient with this, Then it is only I myself hindering Involvement in the cause of merit.

Dorje Tenzin, I'm so very glad to finally have a representative of formal Tibetan school on this site. Do you mind if I ask you to please fill your StackExchange profile with some info about yourself and your tradition?

Dear Andrei Volkov, thank you for your interest. On your request, I filled my profile.

To add few points to nemo's answer which is basically correct.
As per Chogyam Trungpa Rinpoche, the Buddhist definition of patience (ksanti) is freedom from aversion (dosa). Such aversion often manifests as one of: aggression, irritation, anger, resentment or depression.
An important idea to get is that fully developed transcendental patience (kshanti-paramita) is not based on suppression! It is not forbearing an annoying person or cold winter or the sounds distracting one from the sitting meditation while silently hating it!
Instead, transcendental patience is based on A) realizing the nature of the problem, B) realizing the nature of one's own aversion and C) realizing the constructive attitude leading towards the Higher Goal.

Realizing the nature of the problem is to see and acknowledge the objective causes of the situation. It may be atmospheric conditions, or other person's pain and confusion due to their own childhood living conditions or whatever. By analyzing and tracing back the sources of the issue you no longer see it at the superficial level, which removes a lot of negativity.
Realizing the nature of one's own aversion is to see and acknowledge own's own preconceptions / stereotypes / attachments that contribute to superficial vision of the situation.
Realizing the constructive attitude is to abandon the reactive response to the unacceptable situation, a response designed to protect one's ego and to uphold one's preconceptions - and instead to chose a course of action that optimizes the future outcomes. If conflict with a person is brewing, would passive-aggressive behavior and patronizing improve the situation, or should you change the strategy and try and make the person your ally - or on the contrary to come out openly with your issue. Sometimes (usually when you are in the role of a parent or teacher or manager) manifesting aggression is more constructive than suppressing the resentment. What's important to keep in mind is the Higher Goal. Instead of getting obsessed with an obstacle causing our irritation, we can almost always find an alternative way to work around it and get back on the path to the goal.

The point here is that transcendental patience is not a patience of a victim, it is patience of a master.
Development of patience-paramita is related to Buddhist training point known as "taking all bad circumstances onto the path". Instead of mentally labeling problems as punishment or aggression coming from the world, we mentally thank the world for the opportunity to improve our practice.
A very specific part of this is the practice of taking all the blame upon oneself. When we meet any unacceptable situation with a very bold and ego-less step of taking all the blame, taking responsibility unconditionally, regardless of who is right -- something magical begins to happen. The bubble of ego energy that was getting pumped by the aversion bursts and a true opening can take place, a true awakening and opening of the eyes. This opening is the paramita of patience.

"Patience" can be seen as passive, or active.  If I simply endure whatever occurs (even, perhaps, if I am aware of it, and of my experience of it), I am patient in a passive sense; but to "suffer" can also have more active senses.
Perhaps the added element would then be intentionality, or an aligning of my will with, the fact of my experience. At which point, the will can evaporate out of the interaction, leaving reality.
Acceptance, to put it more simply.  A whole-hearted engagement with my experience of suffering (or of whatever I am experiencing, including "good" experiences), unconditioned by judgement, reservations, expectations - by "thinking", as it works against reality - and perhaps grounded on the assurance that this, and nothing else, is the reality of the universe for me, now, and to  imagine / crave / hope for / plan for / long for anything other than what my experience is, here and now, is a form of self-annihilation.


text:
Celibacy - why not?
In Buddhist forums where I had asked this question, there are always "advanced" practitioners who say that there is no benefit for a lay person to be celibate. So every time i read that, I stop trying to be celibate, although I see some benefit from it (though I'm not sure of it).
I would like to hear the opinions of different users on this subject. Are there any benefits from practising celibacy or is it not useful? (I mean full time celibacy, not only on retreats or at Uposatha days)
If the answer is that it's useful, then why is it not more recommended in Theravada Buddhism? Also monks do not recommend celibacy to lay people outside of Uposatha and retreats. Why not?
I am mainly interested in the view of Theravada Buddhism, but I appreciate all views on this subject also from other sects of Buddhism and other religions like Hinduism etc. All answers are more than welcome. Please specify which tradition of Buddhism or Hinduism that you are basing your answer on.

you may be interested in reading this recent question : In what way(s) does engaging in sexual activity affect meditation practice?

Buddha was notorious for only saying things that no one (ethical and reasonable person) could disagree with. His doctrine of Sat Dharma (true/eternal way/dharma) was specifically designed to be irrefutable. In order to be irrefutable it had to be always true and win/win. In order to be always true and win/win it had to be well-specified. Well-specified means specified as broadly or as precisely as needed, worded very carefully. (This is portrayed in many places in Pali Canon but is subtle enough that I can't easily search for references.)
In this case we can imagine that to require the laity to be celibate would be rather "refutable":
Remember that the goal of laity is good happy living, not nirvana. While some of the Buddhist practices lead to both happy life and nirvana, some like celibacy are not conducive to happy living in the long run. Indeed, imagine if Buddha said "everyone should abstain from sex". If everyone agreed, the civilization would die out. So much for the happy living!
On the other hand, if everyone disagreed, Buddha would likely lose some of his credibility. If some people agreed and some disagreed, there would be a lot of arguments in society.
However you look at it, it just doesn't make sense to advise the laity to be celibate.
Are there any benefits from practicing celibacy or is it not useful?
Now, of course if celibacy was not useful for attaining the goal of spiritual life, Buddha would not recommend it to his serious students.
From my teachers, the main reasons for celibacy are:

Sexual lust is one of the strongest defilements, so being able to abstain is a huge step towards mental/emotional sobriety.
Sexual relationships bring up lots of people problems like attachments, conflicts, breakups etc.
Orgasm is a giant waste of psychosomatic energy.

In Vajrayana, there are also reasons against celibacy, particularly if you are the kind of person who has fears around sexuality and intimacy. If that is the case, your teacher may prescribe sex as a kind of spiritual practice.

"If everyone agreed, the civilization would die out. So much for the happy living! " Why? There would be nobody to suffer, isn't it exactly what Buddhists want to achieve?

Buddhists for themselves yes, but not laity for themselves! Buddhism is not prescriptive, as in "you ought to do X", it is methodological, as in "if you want to achieve X you have to do Y".

Any method which can help layman to be celibate? Thanks

From non-celibacy arises suffering. How could that be?
From non-celibacy arises jealousy, fear, hatred, tension and all the worries and consequences that arise from lust.

I would like to hear the opinions of different users on this subject. Are there any benefits from practising celibacy or is it not useful? (I mean full time celibacy, not only on retreats or at Uposatha days)

Benefits are less jealousy, fear, hatred, tension and all the worries and consequences that arise from lust. You should self asses yourself and determine if your lust is hindering your spiritual advancement.
The more lust there is, the more the mind is cluttered, clouded and confused, and the harder it is for that mind to concentrate and attain higher concentration levels where the ultimate truth could be experienced.

If the answer is that it's useful, then why is it not more recommended in Theravada Buddhism? Also monks do not recommend celibacy to lay people outside of Uposatha and retreats. Why not?

There is no good in forcing lay-people to become celibate. They should gradually work towards detachment from their desires (including lust), and celibacy will follow naturally.
My answer is not based on any tradition of Buddhism or Hinduism. It's my personal experience gained from meditation.

In the cycle of existence, there is no satisfaction in sensual desires, which are like honey on a razor's edge. ~Shantideva, Bodhisattvacharyavittara

Celibacy is certainly good but it isn't easy in the modern world, like being healthy is good, but being healthy in the middle of a pandemic isn't easy.
We live in a hyper-sexualized world today where even cars need sexy lines to sell.
There's a Vietnamese Zen saying that a tiger is safe in the forest, but at risk when he enters the village. Similarly, a Yogi is safest when he's away from sense triggers: sex, money, luxurious food, liquor, bad companions - all of which can bring up base desires and aversions.
The Thai forest monks, Himalayan tapas-yogis of Hinduism, desert fathers of Christianity all sought relief from such triggers by avoiding worldly contact.

In the Buddha's age exposure to sense desires would have been fewer, and solitude would have been easier to find (no TV, social media, electricity, mobile phones, traffic). Yet, he recommended seclusion, because sense desire is a furious fire that's hard to put out once started. It's basic mental hygiene and sanitisation.
If we are going to perform surgery on our minds (i.e.) serious meditation, then we don't want to do it in an unhygienic condition. Retreats are designed to offer hygienic conditions for the mind.
The practice of "right livelihood" ensures there are few opportunities to tells lies, act dishonestly or violently. Similarly, one must find a place to live where sexual energy is not omnipresent to learn to be celibate, otherwise frustration will set in.
Such frustration arising from ignorance of causes and conditions might then turn practitioners away from the whole of Dhamma. Some teachers might have this in mind while recommending celibacy only on retreats.
It's not a universal rule, to a determined one no wall is too high, but parents often tell their little kids to not climb tall structures until they are older.
I am independent of any sectarian affiliation, so my answer draws from my interpretation of personal experience guided by Theravada, Zen, Hinduism (several kinds), Christianity, modern science and humanities etc.

I find a very practical approach of Bill Hamilton in Saints & Psychopaths useful (pg. 95; emphasis mine, as it concerns this topic the most; the rest is for context):

The purpose of Buddhist precepts is pragmatic in that they are
  directed at achieving a quiet mind. If your sex life is causing mental
  agitation in your meditations, you should change your behavior. While
  on retreat, you should avoid letting your senses wander to sexually
  stimulating objects and direct your attention to meditation objects.
  In the daily life situation this may result in becoming a horny, neurotic celibate which may not be good for your practice. It is
  better to be simple and direct in your views, instead of getting
  involved in elaborate logical reasoning about right and wrong.

The Dhammacakkappavattana Sutta states:

"There are these two extremes that are not to be indulged in by one
  who has gone forth. Which two? That which is devoted to sensual
  pleasure with reference to sensual objects: base, vulgar, common,
  ignoble, unprofitable; and that which is devoted to self-affliction:
  painful, ignoble, unprofitable. Avoiding both of these extremes, the
  middle way realized by the Tathagata — producing vision, producing
  knowledge — leads to calm, to direct knowledge, to self-awakening, to
  Unbinding.
"And what is the middle way realized by the Tathagata that — producing
  vision, producing knowledge — leads to calm, to direct knowledge, to
  self-awakening, to Unbinding? Precisely this Noble Eightfold Path:
  right view, right resolve, right speech, right action, right
  livelihood, right effort, right mindfulness, right concentration.

Here we have the teaching of the middle path between two extremes (sensual indulgence and austerities).
So, based on this, the following were developed (Theravada tradition) for full time practice:

Fully ordained monks (bhikkhu) - 227 Patimokkha rules and Vinaya
Novice monks (samanera) - Ten Precepts
"Homeless ones" (anagarika) - Eight Precepts
Lay persons - Five Precepts (full time), eight precepts (Uposatha, retreats)

And for all categories, the Noble Eightfold Path.
The main idea is that these four different levels of practice were created or evolved, based on the four different types of needs of training of practitioners, which fall under the middle path of the Noble Eightfold Path.
It can be stretched even further in both directions, for e.g.

Even then, the five precepts are not mandatory to lay persons. It is a voluntary training that lay persons can choose to undergo or not. However, it is very highly recommended to serious Buddhists.
Bhikkhus can voluntarily follow the 13 ascetic practices known as dhutanga which are even more strict than usual.

So, while part time celibacy may be useful for lay persons according to this question, it is not a good idea to stray from the middle path by suggesting that lay persons adopt full time celibacy. Lay persons can always choose the other three paths (anagarika, samanera, bhikkhu) if they want to deepen their training.

i ask about celibacy but i allready keep the 5 precepts - i dont see a problem to be an anagarika    -
now i told this new info does your view change about it ? is being an anagarika reccomended ?   i will just avoid listening to music and other forms of entertainment and eat before 12:00
why does it matter if i take all the 8 precepts or just 5 precepts + celibacy - what is the big difference when its called anagarika ? why it matters so much - that its better than being an "upgraded" lay follower ?

Though it is not clearly specified, anagarika apparently do not have much personal possessions and probably spend most of their time in practice and service. So, I don't think they are employed full time in worldly jobs either, and develop careers. But like I said, it doesn't seem to be clearly specified. It seems to be like a "flexible monk wannabe" role.

Of course, even as a lay person, you can also practise celibacy full time. It's just that it is not to be recommended to the masses. Not to be recommended to other people.

Many of the rules of the vinaya and precepts are clearly concerned with the organization of a sangha in the sense of a band of people who take care of each other. Should any member get married and have kids, this would divert that member's attention to his biological family.  This particular concern is irrelevant for a lay Buddhist.
The next point is that a householder's time and attention are divided, but a monk attention can be devoted to the dharma full time. This concern would be true for a lay or monastic Buddhist. The Upaseka Sutra ends each chapter with a reminder of how much more difficult it is to gain enlightenment as a lay follower.
Finally, from the traditional life story of the Buddha, householding is seen as part of the problem with the world. Householding (and by extension, having girlfriends, and all that goes with that) has certain complications (and a less pessimistic person would say certain benefits). This would be true for the lay and monastic Buddhist.
The old monastic system relies on the existence of laity-- the monks were forbidden by the vinaya from engaging in any self sufficient economic activity. So I imagine it was in the interest of the sangha to encourage the laity to farm, do business and in those days, farming meant also having a family as reliable source of labor.
Nowadays, economic conditions are entirely different.  Serious Buddhist can take long vacations from work every year to sit and meditate, with a college education they can participate in reading and writing of Buddhist philosopy. This is a very modern phenomenon and the traditional schools haven't really decided what to do with the "very serious but not monastic Buddhist". The closest traditional thing I can find is from the Mahayana Tradition where lay Buddhist could take the Upaseka and Bodhisattva precepts as a way of being very serious about Buddhism while still being a lay householder.

Actually the "very serious but not monastic Buddhists" have been around since the Buddha's time. In DN 29 ( https://suttacentral.net/en/dn29 ) the Buddha talked about a broad spectrum of the spiritual community from elder monks to celibate and non-celibate householders, lit. "householders of the white robe, holy livers" and "householders enjoying sensual pleasures".

Are there any benefits from practising celibacy or is it not useful? (I mean full time celibacy, not only on retreats or at Uposatha
  days)

Celibacy is freedom & also saves lots of money because when sexual desire ends, most other forms of sensual attraction & desire also end. No more buying expensive flowers & restaurant food.

If the answer is that it's useful, then why is it not more recommended in Theravada Buddhism? Also monks do not recommend celibacy to lay people outside of Uposatha and retreats. Why not?

It is not recommended because monks do not want to impose heavy burdensome expectations upon lay people.
When a person becomes a Theravada monk, they are indoctrinated in a very dualistic manner that the lives of monks & laypeople are completely different.

I am mainly interested in the view of Theravada Buddhism, but I appreciate all views on this subject also from other sects of Buddhism
  and other religions like Hinduism etc.

A very good answer is found in the Christian Bible:

The disciples said to him, “If this is the situation between a husband and wife, it is better not to marry.”
"Not everyone can accept this word,” Jesus answered, “but only those to whom it has been given".
Matthew 19:12
It is good for a man not to have sexual relations with a woman.... I wish that all of you were as I am. But each of you has your own gift
  from God; one has this gift, another has that... Now to the unmarried
  and the widows I say: It is good for them to stay unmarried, as I do.
  But if they cannot control themselves, they should marry, for it is
  better to marry than to burn with passion.
1 Corinthians 7

The benefits of celibacy are that you're not expending energy.  The justification for using it is either to 1) create life or 2) to create more love than you're expending.
If you can't or don't do either, for whatever reason, it IS better to remain celibate and save your life force.

It's hard to answer this question directly. You're asking why "advanced practitioners" said what they did, in reply to whatever your question was. Perhaps you should ask them!

From a Theravada perspective, there's a book titled The Buddha's Teachings on Prosperity, which is written by a Bhikkhu from Sri Lanka (at least in part for an American audience).
Its premise is that there are meant to be two societies, i.e. lay society and sangha; that these support each other, and so on.
I recommend it. It identifies suttas (there are many) intended for lay people. If nothing else you might find it helpful to read a positive/prescriptive message, e.g., "If celibacy isn't recommended then what is recommended?"

I cant ask them - if i could have i wouldnt have asked here...

Maybe you shouldn't, necessarily, believe them. You wrote, "So every time i read that, I stop trying to be celibate, although I see some benefit from it" ... I think you're supposed to note, at least to some extent, your own experience.

That is true that is why bottom line i do try this and why i asked this question here to hear more views on this


text:
Is the sound an object or an experience?
For experience to occur you need to have an object(sound or speaking man?), a sense faculty(ear) and a sense consciousness(hearing). If a sound is an object, what is then an experience?

According to MN 18:

Dependent on the ear and sounds, ear-consciousness arises.

Definitions

Ear = the sensitive part of the ear that is able to detect sound
Sound = vibration
Ear-consciousness = the Thought Moment that is aware of sound

This is the sensing process by which a vibration is registered by the mind. Later thinking processes may combine multiple vibrations into a syllable and multiple syllables into a word. Later thinking processes may recognize the word and start associating a meaning with the word.
So when you "hear a person speaking", it is actually a large number of sensing processes followed by a much larger number of thinking processes.
The sensing process is "impersonal"; a mechanical process that would be the same for you and for me. The thinking process is "personal"; your thinking sequence and my thinking sequence will be different based on our individual past experiences (for example, if the word is in a language that you know but I do not know, then obviously your associations will be very different from my associations).

From a Tibetan point of view, it concerns what is referred to as Lo Rig (epistemology), 'Awarenesses and knowers' and 'Minds and mental factors'.
Strictly speaking, experience refers to a characteristic of the 'feeling' mental factor. As in Yeshe Gyeltsen's Necklace for Those of Clear Awareness:

QUESTION: What is the defining characteristic of feeling?
  RESPONSE: It
  has the characteristic of experience through the entity of experience,
  it experiences individually the fruitional results of virtuous and
  non-virtuous actions.

Also, from a Sautrantika point of view (Dignaga's Compendium of Valid Cognition, Dharmakirti's commentary,
Purbuchok's Explanation), sound would be the observed object condition, casting its aspects onto the ear-consciousness apprehending it.
Sound is an object of a sense direct perceiver (ear-consciousness). As such, it is form. Sautrantika characterize form as being “atomically established”, contrary to Yogacara.
Thus, if you states that “sound is [an] experience”, if absurdly follows that “experience is form... experience is atomically established... experience is an object of a sense direct perceiver... so experience must be seen, heard, tasted, touched or smelled”. As it is not so, experience is not form, and sound is not an experience [because sound is form].
One might assert that the ear-consciousness, “perceives its object” or “experiences its object”, and in this way “has an experience” or “possesses an experience” and is an object-possessor of both the sound and the experience. It would probably not stand in debate, but we might present things like that generally.

"Sound" is an oject and an experience. "Sound" is a certain experience  that is an object. Anything or anybody can be the object of experience. The object is the part of the experience that is being experienced.
The subject is what is experiencing or witnessing the object of experience.

Sound is an object.
When the mind directs itself towards an object, that is the experience. What is the experience? Whatever arises when the mind directs itself towards an object. What arises? Senses and consciousness. Then, thoughts.

I agree with this simple answer. Some of the others seemed to go round in circles. I would say that if a physical cause strikes an inanimate object, that is an 'event'. If it occurs to a simple living thing, like a fish, that is a 'happening'. To something with self-awareness like a dog, it is an 'experience', and people ascribe 'meaning' to things that occur. Sound would have a different context in each of these cases. For an anesthetized person, sound may well cause no mental effect at all, so it would be an event or happening rather than an experience.

As Master Dogen said:

The ways of viewing mountains and waters are different depending upon what kind of beings we are. There are some beings that view water as a jewel. However, this does not mean that they view a jewel [for human beings] as water. How do we see what they view as water? What they see as a jewel is what we see as water. Some beings see water as wondrous flowers. But they do not use flowers [for human beings] as water. Hungry ghosts view water as raging fire or as pus and blood. Dragons and fish view it as a palace or a lofty building. [Some beings] see it as the seven treasures or the mani jewel. [Others] see it as a forest or walls, or as the dharma nature of immaculate liberation, or as the true human body, or as body as the form and mind as the nature. Human beings view it as water. And these [different ways of viewing] are the conditions under which [water] is killed or given life.

Some beings (dogs) perceive vibrations of higher frequencies that other beings do not perceive as sound at all. For some (tiny) beings, the vibrations we perceive as sound are perceived as physical movement. Some beings, like bats, perceive sound as vision.
There is nothing inherent in the vibration that makes it sound. Very slow vibrations (< 1Hz) are not perceived as sound at all. Vibration itself is not an entity either - it is a process of cause-and-effect propagating through various media.
The point here is that phenomenal reality is projection, an interpretation that depends on the point of view. Because experienced reality is projection, we can control it by controlling our point of view.


text:
Is aversion toward unwholesome deeds normal?
The more we do wholesome deeds the more the inclination to do more good deeds and avoid unwholesome deeds.
Wholesome deeds are encouraged but not unwholesome deeds.
We need to keep doing wholesome actions, (attach to them) make them become our habit, the same thing with unwholesome actions if we do not want suffering.
So is avoiding unwholesome actions the same thing as aversion? It's said even Sakka Devaraja is not free from aversion. Is it normal and necessary to have or cultivate aversion toward unwholesome actions than toward wholesome actions?

Hmm, I don't know, but what I read from research and from reports about native people it seems to me that not only avoiding of unwholesome deeds is completely natural, but even an aversion against such deeds. However -whether it is normal and necessary to cultivate aversion : well, personally I don't believe that. And I think in Buddhism we have the tendency (and also the explicite will) to reduce/to emancipate from the emotional affect of aversion in favor of detachement and equianimity.

On the initial stages, cultivation of aversion towards unwholesome is normal, one might say even critical.
There are numerous examples in the Pali Canon suggesting cultivation of aversion to the unwholesome, many of them revolving around cultivation of disgust with the body as an antidote to lust:

Since your perception is distorted,
  Your heart with passion is aflame.
  The marks of beauty should you shun,
  Bound up with lustful longing and desire.
Your mind, one-pointed and collected,
In seeing foulness should be cultivated.
  With mindfulness directed on the body,
Dwell often in disgust concerning it.

But also more generally:

Having done a mental action, you should reflect on it: 'This mental action I have done — did it lead to self-affliction, to the affliction of others, or to both? Was it an unskillful mental action, with painful consequences, painful results?' If, on reflection, you know that it led to self-affliction, to the affliction of others, or to both; it was an unskillful mental action with painful consequences, painful results, then you should feel distressed, ashamed, & disgusted with it. Feeling distressed, ashamed, & disgusted with it, you should exercise restraint in the future.

On the middling stages, once the separation from the coarse unwholesome qualities (habitual tendencies) has been accomplished, cultivation of aversion is substituted for cultivation of joy:

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; just so, the monk permeates, suffuses and fills this very body with the rapture and pleasure born of [directed thought and evaluation on one's successful] withdrawal [from sensuality, from unskillful qualities].

And then on the advanced stages, cultivation of joy (and then equanimity) is superseded by cultivation of disenchantment:

... that he keep cultivating disenchantment with regard to form, that he keep cultivating disenchantment with regard to feeling, that he keep cultivating disenchantment with regard to perception, that he keep cultivating disenchantment with regard to fabrications, that he keep cultivating disenchantment with regard to consciousness.

From Kimattha Sutta:

Ananda: "And what is the purpose of disenchantment? What is its reward?"
Buddha: "Disenchantment has dispassion as its purpose, dispassion as its reward."
A: "And what is the purpose of dispassion? What is its reward?"
B: "Dispassion has knowledge & vision of release as its purpose, knowledge & vision of release as its reward.
Ananda, skillful virtues have freedom from remorse as their purpose, freedom from remorse as their reward. Freedom from remorse has joy as its purpose, joy as its reward. Joy has rapture as its purpose, rapture as its reward. Rapture has serenity as its purpose, serenity as its reward. Serenity has pleasure as its purpose, pleasure as its reward. Pleasure has concentration as its purpose, concentration as its reward. Concentration has knowledge & vision of things as they actually are as its purpose, knowledge & vision of things as they actually are as its reward. Knowledge & vision of things as they actually are has disenchantment as its purpose, disenchantment as its reward. Disenchantment has dispassion as its purpose, dispassion as its reward. Dispassion has knowledge & vision of release as its purpose, knowledge & vision of release as its reward. In this way, Ananda, skillful virtues lead step-by-step to the consummation of arahantship.

These disenchantment and dispassion born of "the knowledge & vision of things as they actually are" is the very end of the path:

the Tathagata — the worthy one, the rightly self-awakened one, who from disenchantment with form, from dispassion, from cessation, from lack of clinging (for form and the rest of the five skandhas) is released — is termed 'rightly self-awakened.' And a discernment-released monk — who from disenchantment with form, from dispassion, from cessation, from lack of clinging (for form etc.) is released — is termed 'discernment-released.

Once "the knowledge & vision of release" has been achieved, there is no cultivation at all:

One who is dependent has wavering. One who is independent has no wavering. There being no wavering, there is calm. There being calm, there is no yearning. There being no yearning, there is no coming or going. There being no coming or going, there is no passing away or arising. There being no passing away or arising, there is neither a here nor a there nor a between-the-two. This, just this, is the end of stress.

And finally, total unbinding:

Any form (and the rest of five skandhas) by which one describing the Tathagata would describe him: That the Tathagata has abandoned, its root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Freed from the classification (of form etc.), Vaccha, the Tathagata is deep, boundless, hard to fathom, like the sea.

I think aversion and a sense of disgust are not only for initial stages.
Knowledge of disgust (nibbida-ñana) is one of the knowledge among other knowledges such as knowledge of equanimity, etc.:
Seeing thus the misery in conditioned things (formations), his mind finds no delight in those miserable things but is entirely disgusted with them...

...Even if he directs his thought to the happiest sort of life and existence, or to the most pleasant and desirable objects, his mind will not take delight in them, will find no satisfaction in them...
(http://www.accesstoinsight.org/lib/authors/mahasi/progress.html#ch6.8)

In the big picture Knowledge of Desire for Deliverance and Knowledge of Disgust - are still rather initial stages.

Abiding amidst a multitude of mental afflictions, one should be vigorous in a thousand ways and unconquerable by the hosts of mental afflictions, like a lion by a herd of deer. ~Shantideva

Fear of the unholy born of an enthusiasm for virtue is justified. Otherwise, we would be like the buffalo that falls asleep among butchers.
However, it should not turn into aversion of those still afflicted by it.
The great and noble Shantideva, writes frequently on this subject in his guide to the Bodhisattva way of life ("Bodhisattvacharyavittara") -especially in the chapter on perfection of zeal.
[I just noticed the Theravada tag, apologies]
Rather than phrase your position as aversion to the unwholesome, it might be better to describe it as wise pursuit of the Dhamma.
Please see sections 7,8 of MN 139 Araṇavibhanga Sutta - The Exposition of Non-Conflict - one must neither extol nor disparage but teach only the Dhamma.


text:
How careful must monks be about their use of water?
According to this list of the Patimokkha Rules,

Using water, or getting others to use it, knowing that it contains
living beings that will die from that use, is a pācittiya offense. (Pc
62)

So, does this refer even to microbes in water? If so, how would a monk boil water for food and/or tea without committing an offense? Or even drink water without committing an offense?

The common answer is that microbes and smaller life forms are exempt from Buddhist classifications of sentient life forms.
I'd like to make a more nuanced point.
At the time it was written, this rule probably had mosquito larvae in mind.
Practically speaking, even abiding by this ontological line in the sand is difficult if the only source of water is infested with mosquitoes or the numerous life forms of tropical Asia. Most monasteries therefore often rely on lay helpers to perform water boiling and several other life tasks that might break the vinaya if a monk were to perform it.
To an outside observer this can appear dogmatic or a farce, and depending on the individuals involved sometimes it is so, but not always so. It aids in cultivation of zeal, and respect for the expectation placed on the monastic.
The whole of the spiritual life is doing one's best in trying conditions.
The Bodhisattva vows reflect this,

However innumerable sentient beings are, I vow to save them.
  However
  inexhaustible the defilements are, I vow to extinguish them.
  However
  immeasurable the dharmas are, I vow to master them.
  However
  incomparable enlightenment is, I vow to attain it.

In traditional Asian cultures the laity often view these small inconveniences the monks undertake, like waiting around for a lay person to give them potable water as resolve strengthening sacrifices the monks make in the pursuit of the holy life. It also gives the laity a meritorious role in the life of the monastic, and keeps them inter-dependant.
This can also be a very instructive exercise in patience and virtue when applied correctly.
For example, if a novice monk is alone and thirsty in the forest, it must be very tempting to drink a cup of water after pushing aside the larvae on top of the forest pool.
This simple physical act is not so simple in the mind of a monk, given the vinaya. It leads him weigh his life against that of the organisms in the water.
In this way the monk comes to directly feel the enormous burden human life places on the planet, and the numerous beings and life forms including plants that we inadvertently or unavoidably destroy or disturb in order to live.
Thus chastised, the monk will truly consider the cup of water a precious cup of life, and endeavour to uphold the ideals of the holy life so stiffly purchased.
The rules are meant to aid the holy life, not become obstacles or rituals. The Buddha reveals this pragmatic distinction here:

AN 3.83: Vajjiputta Sutta — The Vajjian Monk {A i 230; Thai 3.85} [Thanissaro]. A monk who is having difficulty following all the Pāṭimokkha training rules can boil them all down to these three: the training in heightened virtue, the training in heightened mind, the training in heightened discernment.


text:
Which aggregate appears before...perception or feeling?
Do you first feel the touch (pleasant, unpleasant or indifferent) or do you first identify it? Or is it all at the same time?
Is the order of aggregates always the same for all the senses?

The same terms are often used at a "microscopic-view" level, at a "day-to-day-view" level and at a "macro-view". This leads to a lot of confusion.
According to the Abhidhamma, feeling / sensation / experience (vedanā) and perception / recognition (saññā) arise simultaneously in every Thought Moment (cittakhaṇa). The Abhidhamma looks at things at a "microscopic-view" level.
The Suttas (MN 18 is an example) describe how sensing leads to thinking. At this "day-to-day-view" level, vedanā (part of the sensing process that is the same for everybody) leads to saññā (start of the thinking process; a subjective process influenced by latent defilements leading to distortion of perception).
The aggregates are a way of describing what constitutes a being (what a being is or what a being does). This is the "macro-view" level that describes the nature of existence. The aggregates do not have a sequence (they are not describing a process), so vedanākkhanda (feeling aggregate) and saññākkhanda (perception aggregate) arise simultaneously.

Are you saying that 'sañña' is thinking?

"Saññā" is translated as perception. MN 18 gives a progression: contact -> feeling -> perception -> thinking about. The initial part of this progression is impersonal (with contact as a condition, feeling arises). The  subsequent part of this progression is personal (what one feels that one perceives, what one perceives that one thinks about (vitakka) ). In this Sutta, "saññā" is the stage at which distortions (saññāvipallāsa) are introduced. I described the initial impersonal part as "the sensing process" and I described the subsequent personal part as "the thinking process".

Is this correct....feeling is a characteristic of an object and we all get the same feeling from the same object.

Feeling is a characteristic of the mind, not of the object. At a primitive level, seeing, hearing, smelling and tasting all arise with neutral feeling while touch can be either pleasant or painful (this primitive feeling is impersonal). The mind's reaction to the object can be associated with pleasant, neutral or painful feeling (the reaction and its feeling is personal).

Is this question reasonable....do we have any control over 5 aggregates?

The five aggregates are all natural processes. There is no "self" to have control ("self" is an illusion).

So, what you said that the thinking process is personal, is just an illusion?

Sensing process depends on a working eye and an external object (these are impersonal; same for you and me). Thinking process is influenced by defilements, paramis, habits, accumulations, etc. (these are personal; different for you and for me). Both sensing process and thinking process are natural processes, without a self. There is seeing without a seer, thinking without a thinker.

'Feeling' and 'discrimination' are not only aggregates, they are first and foremost mental factors. As such, they accompany main minds with which they share similarities.
According to Tibetan Lo Rig (Mind and  mental factors), the five omnipresent mental factors (intention, attention, contact, feeling, discrimination) are so called because there is not one main mind (such as an instance of an eye-consciousness, ear-consciousness... mental consciousness) that is not associated with the five.
For instance... an eye-consciousness apprehending blue is necessarily accompanied with the five omnipresent mental factors. The main mind and the five share five similarities and are one entity:

1. Time (they arise, abide, cease simultaneously),
2. object of observation,
3. Subjective aspect
4. Support
5. Substance
Yeshe Gyeltsen's Necklace for Those of Clear Awareness writes:

QUALM: Are a mind and its mental factors one entity or different
  entities? RESPONSE: The Treasury of Knowledge (stanza 2.23) says:
  A mind and its mental factors are definitely simultaneous.

If you want more on this topic such as exposed by Tibetan Geluk scholars, I suggest The Mind and Its Functions, by Geshe Rabten.

Is this "feeling" that you call mental factor the same as i am thinking...sensation/feeling/vedana?

Yes, it is the same, vedana. It is "one of the seven omnipresent mental factors in the Theravāda Abhidharma", "one of the five omnipresent mental factors in the Mahāyāna Abhidharma", "one of the five skandas in both Theravāda and Mahāyāna traditions". This is why I said "they are not only aggregates, they are mental factors". Because if you understand the relationship a mental factors has with the main mind it is in the retinue of, you will have an answer to your initial question.


text:
Difference between "Becoming" and Anicca
To preface my question, I will add that I am very unfamiliar with Buddhism.  My knowledge consists of some historical anecdotes, and I also know some of the basic tenets.
That being said, is there a difference between "Becoming" and Anicca/Impermanence?  The former is a philosophy that originated in the West, and is nearly as old as Buddhism itself.  Are they essentially the same thing, but stated differently?  Or am I overlooking a huge fundamental difference which delineates the two?

I think, How to become anything is relate to "Becoming"(patichcha) and Anicca/Impermanence is everything's "Behave".

To answer this question, must someone know what western philosophers have written about "becoming"; and might you therefore get a more-informed answer on Philosophy.SE instead? Are you able to rephrase this question, either to supply in your question a stand-alone definition of "becoming", or to ask a question that's specifically/directly about anicca without asking for it to be compared with "becoming"? I assume that to some extent you're asking how it compares with your (personal) preexisting understanding of "becoming", but few people here can guess what your understanding of becoming is.

In Buddhism, anicca might be a axiom or observation from which logical consequences, e.g. anatta and some types of dukkha, and consequently recommendations like non-attachment and/or equanimity, are derived. I don't know what the consequences/conclusions are of "the 'becoming' philosophy".

I suppose I could migrate this question to Philosophy.SE, although I am specifically interested in the Buddhist perspective (granted, there is no such thing as a universal "Buddhist perspective," but I doubt that this issue is a hotly debated topic without some sort of consensus).

Yes, however your Wikipedia reference quoted Nietzsche, and that's why I suggested Philosophy.SE as an alternative (because IMO relatively few people here know what Nietzsche wrote about Becoming, and the specific subject of this question is "a comparison with Becoming").

My answer assumes you are specifically referring to the ideas of Heraclitus (the originator of Western ideas of Becoming). There are some interesting correlates between the origins of the Western idea of flux or becoming which was first articulated by the Greek philosopher Heraclitus and the Buddha's idea of Impermanence (Anicca). For Heraclitus ...

Everything flows and nothing stays.
  Everything flows and nothing abides.
  Everything gives way and nothing stays fixed.
  Everything flows; nothing remains.
  All is flux, nothing is stationary.
  All is flux, nothing stays still.
  All flows, nothing stays.
No man ever steps in the same river twice.

There is subtle distinction between the Buddha and Heraclitus on the mechanism of change. For Buddha the process of change is driven by dependent origination - the idea that every phenomena exists entirely dependent on the causes that proceed it. Heraclitus had more of a transformative concept of change - that phenomena are the result of the tension and flux of the opposites. The end result is the same however - constant change.
So essentially there is very little difference between the Buddha's concept of  Annica and Heraclitus's ideas of Becoming.
Other interesting similarities between Heraclitus and Buddha:

Both were born into aristocratic families and were first in line to inherit.
Both rejected their inheritance and chose rather to pursue a life of contemplation instead.
Both their philosophies rejected the prevailing eternalism of their respective cultures and chose rather to rely on reason and personal experience to arrive at the truth.
They were fairly contemporaneous with each other - Heraclitus (c. 535 – c. 475 BCE) & Buddha (6th century BCE)

There are many articles and books dedicated to investigating the similarities between the views of Heraclitus and Buddha on the internet. Nice summary here -  Heraclitus and the Buddha on Impermanence

According to Buddhism the "Becoming" is define as create "Nama-Rupa".
The Alternatives for nāmarūpa is saṅkhāra.
All the sankaras are impermanence. That is called Anicca.
saṅkhāra˚;

Definition for the pāli word Santati (line)
  continuity of states rūpa˚ of form;
  causal connection of material things
“Empty is this, bhikkhus—the saṅkhāras—either of soul or of what
  belongs to soul”