"What happens to the five skandhas of a particular person after death?"
As far as I understand the doctrine (since I can't remember my deaths, so far): they cease. Then, new skhandhas appear, conditioned or causally related by the previous ones (no different answer than all the others here, i guess).
"Do they stay together, and cause birth of a one specific person (which can be called rebirth), or do they separate?"
The above seems to suggest skandhas are objects or structures. I think it's troublesome, for example, to understand sañña as a structure, existing or tied to the other four skandhas (which also exists somehow?); in other words, I don't see a sañña structure that "perceives perceptions". This also seems to lead to an Eternalist view.
What the suttas say is that:
Dependent on the eye and forms eye-consciousness arises. The meeting of the three is contact. With contact as condition there is feeling. What one feels, that one perceives.
-- Madhupindika Sutta (Ñānamoli/Bodhi trans), MN 18
Thus, for example, when there is "perception of blue" there is sañña (this is sañña). When there is pleasant feeling, there is vedanā. The pleasant experience itself is vedanā. Observing that feeling cease, is observing vedanā cease.
So, in this perspective, we experience all those "things", moment after moment. They are in the domain of our reality. In our individuality, these experiences are very inter-related and we refer to them to describe ourselves and understand our process of becoming. Through the skhandha point of view, we are taught how to distinguish each discrete real experience and understand it, its kind and its role in that overall process.
Now the pleasant feeling subsides. Right after, to say vedanā is still "there", when it's not, is imagining something else "behind" and declaring it's name is "vedanā".
"If they separate, do they cause births of many different persons?"
Again, that seems to imply that each skandha is an object or individual self on its own. Somewhat related:
Karel Werner (1988) has sought to develop an alternative explanation of how character-continuity is conserved. He takes early Buddhist terms such as vinnana (discernment) and citta as referring, in certain instances, to a "personality structure" with changing contents, but an ongoing identity (p.89). He sees this, in turn, as similar to the Vedic idea of tanū, the "likeness" which when 'filled' with phenomenal elements is a person's 'character', but which momentarily exists without content after death (pp. 78-9). Such an "empty structure" is not an unchanging entity but a "structural continuum which registers and preserves ... the imprints of past experiences, volitions and capabilities" (p.79). The "personality structure" of such a model, though, seems indistinguishable from the 'person' of the Personalists (Pudgalavāda), which has already been shown to lack any foundation in the 'early Sutta' world-view (Paras.l.36-42).
-- Peter Harvey, The Selfless Mind, pg 73
Now, to use Sue Hamilton's words:
[the Buddha teaches] that the analysis of the human being into five khandhas is not an analysis of what the human being consists of, but of those processes and events with which one is constituted.
-- Identity And Experience: The Constitution of the Human Being According to Early Buddhism
"EDIT: I didn't get unambiguous answers to my questions, but what Suminda and Prahlad write seems to suggest that all the aggregates arise and pass all the time"
we are all in agreement, it seems.
"but when we look at them at a higher level, we can conclude that four of the aggregates (sensation, perception, mental formations and consciousness) stay together and cause a new birth of one specific person, while the remaining aggregate (rūpa, matter) separates and splits into parts. Can anyone confirm/disconfirm this?"
You seem to have imagined another set of aggregates that do not arise and pass away, but whose parts are permanent and tied together. And you seem to be imagining them existing in this "higher level".
I don't think there is a "higher level" where these imagined 5 aggregates can be actually observed outside imagination. There is no vedanā without an actual feeling. There is also no "same body" persisting through time.
"EDIT 2: I was asked to specify what is exactly meant by "overall structure of the aggregates remain intact", when we know that the aggregates arise and pass all the time. What I mean is the continuity of the illusion called "self". I have the illusion that I'm the same person as a moment ago, even though my body, my feelings, etc. have changed. Does this illusion continue after death?"
I'm not sure I understand. Considering "self" to be what is understood in Buddhism (permanent, eternal, dukkha-free, etc):
Do you mean to ask if you will still believe in a self after death? I don't know.
Or do you mean to ask if sentient beings, after death, are still led to believe in self? According to Buddhism, likely, yes.
Or do you mean to ask if, after death, it will be obvious that the self is an illusion? According to Buddhism, not necessarily.
"According to Buddhist doctrines, is there an illusion of the continuity of consciousness? Of the continuity of feelings? Of the continuity of the body?"
Buddhism acknowledges that people hold to the above beliefs. That is, that people end up believing that consciousness, feelings, body, etc. are permanent. And the Buddha declared these beliefs as misleading.
"I watched Yuttadhammo's video about rebirth and understood that Buddhists believe in perceived continuity of the consciousness after death. Is this a canonical opinion that can be concluded from Buddhist scriptures? How about the continuity of other aggregates?"
I don't know which video you watched, but I would be surprised to know that Yuttadhammo claimed that consciousness (viññāṇa) transmigrates. There is a sutta about a bhikku named Sāti who was debunked by the Buddha for declaring this:
Now on that occasion this pernicious viewpoint had arisen in a bhikkhu named Sāti, son of a fisherman, thus: "As I understand the Dhamma taught by the Blessed One, it is this same consciousness that runs and wanders through the round of rebirths, not another."
[the Blessed One said to him] "Misguided man, to whom have you ever known me to teach the Dhamma in that way? Misguided man, have I not stated in many ways consciousness to be dependently arisen, since without a condition there is no origination of consciousness?
-- Mahatanhasankhaya Sutta (Ñānamoli/Bodhi trans), MN 38
Perhaps in said video, he could have talked about "stream of consciousness", which is not the same thing: in one sutta (and apparently nowhere else?) the respective term, "viññāṇa-sota", appears:
[...] he comes to know the unbroken stream of human consciousness as established both in this world and in the next.
[...] he comes to know the unbroken stream of human consciousness that is not established either in this world or in the next [n: Arahants].
-- Sampasādanīya Sutta (Walshe trans), DN 28
In the notes, we read:
viññāṇa-sota: a rare expression which seems to equate with bhavanga, the (mainly) commentarial term for the 'life-continuum' (Nānamoli). [...]
Another sutta that refers to the role of consciousness in the process of rebirth is:
"Thus kamma is the field, consciousness the seed, and craving the moisture. The consciousness of living beings hindered by ignorance and fettered by craving is established in a [lower/middling/superior] realm. Thus there is the production of renewed becoming in the future."
-- Bhava Sutta, AN 3.76
In general, I understand there is no consensus as to how the process of rebirth happens in detail and later schools seemed to have developed their own theories. A supposedly less polemic link of births is kamma:
"Student, beings are owners of their actions, heirs of their actions, they originate from their actions, are bound to their actions, have their actions as their refuge.
-- Cūlakammavibhanga Sutta (Ñānamoli/Bodhi trans), MN 135