DN 15: Whose or which consciousness enters the mother's womb?

Which consciousness is entering into the mother's womb?

In English, the word consciousness already implies the (mental) self. In the context of this question, the term used in the Buddhist suttas is viññāṇa. It is the viññāṇa that "descends" (figuratively) into the womb.

Here is a (grossly) oversimplified way to understand viññāṇa via the reading (of this answer):-

Story #1
One day, a man decided to venture into a hutama. Not far into the hutama, he heard bunyee and caught a whiff of bauya. He felt jamaya in his being. He looked up and saw hamaza.

Story #2
One day, against the warnings of the villagers, a man insisted to venture into a forest reportedly inhabited by a flesh-eating demon. Not far into the forest, the man heard an unfamiliar sound and caught a whiff of an unfamiliar scent. Fear grew in him. He looked up and saw an unfamiliar figure. He soiled himself where he stood, and on regaining his composure, he ran.

Story #3
One day, upon hearing the presence of a grant-fulfilling tree in a forest, a man decided to seek for it. Not far into the forest, he heard an unfamiliar sound and caught a whiff of an unfamiliar scent. Excitement grew in him. He looked up an saw an unfamiliar figure. He was so happy that he cried.

Story #1 attempted to illustrate the viññāṇa as it arises through the ear, nose, "heart" and eye but stopped at the memory-look-up/comprehension step by using some made-up words. Story #2 and Story #3 attempted to illustrate unpleasantly and pleasantly preconditioned "heart" viññāṇa.

Is it the consciousness of a disembodied being or relinking 
consciousness floating in outer space seeking to be reincarnated?

"Rebirth-linking consciousness" is a misnomer. In the kama-loka, when a being is dying, as the functioning of the physical sense organs deteriorate, so will the functioning of viññāṇa and mana. And when the viññāṇa ceases to function, the bhavanga kicks-in. The Theravadin Abhidhamma texts describe the bhavanga as the absence of mental awareness and comprehension. According to Visuddhimagga, it is this bhavanga that links up the next birth.

Or is it the consciousness of the mother, ...

A free woman can decide whether or not to bear a child. This much as I can say.

Firstly, is there rebirth according to the suttas? (And is it also taught to monks?)

Yes. According to SN 12.19:

Bhikkhus, for the fool, hindered by ignorance and fettered by craving, this body has originated. For the fool that ignorance has not been abandoned and that craving has not been utterly destroyed. For what reason? Because the fool has not lived the holy life for the complete destruction of suffering. Therefore, with the breakup of the body, the fool fares on to another body. Faring on to another body, he is not freed from birth, aging, and death; not freed from sorrow, lamentation, pain, displeasure, and despair; not freed from suffering, I say.

So, the fool gets reborn. But is it the same fool, or a different fool, or it's neither the same fool nor different fool? Which fool is it?

According to MN 38, it's not the same consciousness that gets reborn:

“Exactly so, venerable sir. 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.”

“What is that consciousness, Sāti?”

“Venerable sir, it is that which speaks and feels and experiences here and there the result of good and bad actions.”

“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? But you, misguided man, have misrepresented us by your wrong grasp and injured yourself and stored up much demerit; for this will lead to your harm and suffering for a long time.”

OK. So, if it's not the same consciousness or same individual that gets reborn, then is it a completely different unrelated individual?

Not really - it's not the same individual, yet not completely unrelated individual according to SN 12.17 (but paraphrased here):

Again, when the Buddha was asked by the naked ascetic Kassapa whether suffering was of one's own making or of another's or both or neither, the Buddha replied "Do not put it like that." When asked whether there was no suffering or whether the Buddha neither knew nor saw it, the Buddha replied that there was, and that he both knew and saw it. He then said "Kassapa, if one asserts that 'He who makes (it) feels (it): being one existent from the beginning, his suffering is of his own making,' then one arrives at eternalism. But if one asserts that one makes (it), another feels (it); being one existent crushed out by feeling, his suffering is of another's making,' then one arrives at annihilationism. Instead of resorting to either extreme a Tathaagata teaches the Dhamma by the middle way (by dependent origination)"

I see it as a chain of the succession of related changing processes that are dependently originated based on conditions, which can continue beyond the death of the physical body.

OP: Which consciousness is entering into the mother's womb?

It does not belong to someone. The "descent" of consciousness is not a literal physical movement of something but simply refers to the manifestation of consciousness, just as if I were to say "Buddhism descended into the minds of people".

OP: Is it the consciousness of a disembodied being or relinking consciousness floating in outer space seeking to be reincarnated?

If I were say, "Buddhism descended into the minds of people", you might ask "from where?".

Well, it certainly did not fly in some medium like a particle and enter into people's brains. But it transferred somehow didn't it? When people read Buddhism.SE questions and answers, or someone told them something, it caused them to think about it, and that's how Buddhism descended into their minds.

Similarly, I don't see it as a direct physical transfer of some thing but rather, it's an indirect information transfer of some kind. But I don't know how it works.

This is analogous to quantum entanglement in particle physics. Physicists cannot explain why or how it happens (because there is no physical transfer of information), but they know that it happens.

OP: Or is it the consciousness of the mother, so the dependent arising of the permutations, signs, themes and indicators that lead to her designating her embryo in her womb as "my baby" can occur?

I'll quote the excerpt of DN 15 here:

‘Name and form are conditions for contact’—that’s what I said. And this is a way to understand how this is so. Suppose there were none of the features, attributes, signs, and details by which the category of mental phenomena is found. Would linguistic contact still be found in the category of physical phenomena?” “No, sir.” “Suppose there were none of the features, attributes, signs, and details by which the category of physical phenomena is found. Would impingement contact still be found in the category of mental phenomena?” “No, sir.” “Suppose there were none of the features, attributes, signs, and details by which the categories of mental or physical phenomena are found. Would either linguistic contact or impingement contact still be found?” “No, sir.” “Suppose there were none of the features, attributes, signs, and details by which name and form are found. Would contact still be found?” “No, sir.” “That’s why this is the cause, source, origin, and condition of contact, namely name and form.

Guess what? This seems to fit into Andrei Volkov's description of namarupa in this answer.