What do you mean by “complicated situations”? Is he permitted to have intercourse or not?
Would this be a case of “betrothal not given over for intercourse,” or not?

It has nothing at all to do with betrothal not given over for intercourse, even if intercourse is forbidden.
As for whether intercourse is permitted, that’s an interesting question. Unlike Siamese twins who are merely attached, here he is having intercourse with both of them, because they have one body. I can argue in favor of permitting it, but it really is a question. For example, one could define them as partners in one body, and as is known in the law of partners, when one uses the field, the whole field is considered his. The division is along the time axis. See Ran at the beginning of the chapter “Ha-Shutafin” in Nedarim, at length.

According to that, each one would also be permitted (from a halakhic standpoint at least) to injure herself, even though she thereby causes pain to the other. And one must consider: if one of them is pursuing them both in order to kill them, is it permitted to save the other by injuring one of her limbs, since the pursuer is a murderer? Or is the pursuer only committing suicide, and doing with her own what is hers, having at that moment selected for herself the part of the shared body for her own use—and if so, it would not be relevant to save her by injuring one of her limbs. One must also consider: if we assume that each one can prevent both of them from moving their legs in a given direction, is she nevertheless obligated to allocate time for her sister to use the legs, or is it simply “whoever is stronger prevails”?

Absolutely not. A partner is forbidden to destroy the field during his term of use.

Okay.

The question gets more complicated: is the second sister allowed to marry someone else and have sexual relations? (Again, same case—the second sister started dating someone.)

I don’t think that complicates things. According to what I said, definitely yes.

Right, I only realized that after I sent it.

I just want to make sure: at any given moment, does the whole body essentially belong to the specific sister with whom the husband is having relations?

So a couple more questions came to mind:
1) To whom does the child that is born belong? Do we check by DNA who the father is, and the child belongs entirely only to the relevant sister?

2) Is one sister allowed to use contraception when the other does not want that?

3) If it is known for certain that one sister controls a certain side of the body and the other sister controls the other side, is it permitted only for the relevant sister to cooperate while the other must remain passive? In such a case, would what you said still apply? (Regarding division along the time axis.)

Yes.
1. I think so. Though one could discuss treating the second one as a surrogate, since he ejaculated into her womb. But I am inclined to think she has no status regarding the child.
2. Good question. Presumably not, because she is damaging her body, and as I said, a partner may not damage jointly owned property. On the other hand, not using contraception is damaging for the first one.
3. That is a more complicated situation, because here there is no partnership in everything that is divided along the time axis. This is a partnership in two parts (the second possibility in the Ran there). Except that here the parts are even defined, whereas the Ran there speaks about undefined parts, so this is not really even a partnership but only a kind of adjacent-rights relationship. It could be that in such a situation the whole thing is forbidden, unless we resort here to more complex constructions (such as one sister lending part of the body to the other, and that might depend on the dispute among the medieval authorities as to whether a borrower has ownership or only permission to use it).