Lekarz powiedział, że moje ucho musi odpocząć i że mam zostać w domu.

Because ucho is a neuter noun, and the possessive adjective has to match it in gender, number, and case. So: mój for masculine,
moja for feminine,
moje for neuter.

That is why you get moje ucho.

Because swój refers back to the subject of its own clause. In the clause moje ucho musi odpocząć, the subject is ucho, not I. So swoje would not fit here.

Compare: Muszę oszczędzać swoje ucho = I need to protect my own ear.
Here the subject is I, so swoje works.

After musieć, Polish normally uses an infinitive, so musi odpocząć is the expected structure.

The verb odpocząć is perfective. Here it suggests getting the needed rest, as a complete result. That sounds very natural in this medical context.

The imperfective odpoczywać would emphasize the ongoing process of resting, and in this sentence it sounds less natural.

Mam + infinitive often means something like I am supposed to, I am to, or I have been told to. That is very common when reporting instructions from a doctor, teacher, boss, and so on.

So mam zostać w domu sounds like the doctor’s recommendation or instruction.
Muszę zostać w domu would mean I have to stay at home, which is also possible, but it sounds more like personal necessity than reported instruction.

Zostać here means stay or remain. It suggests not going elsewhere and continuing to be at home. That is why it fits a doctor’s instruction well.

Być w domu simply means to be at home. It describes location, but it does not express the idea of staying there as clearly.

It is the locative case. After w meaning in or at a location, Polish uses the locative:
dom → w domu.

If you were talking about motion toward home, you would usually say do domu, not w domu.

Because Polish often drops subject pronouns when they are not needed. The verb form already shows the person: mam means I have / I am supposed to.

You can say ja mam, but that adds emphasis or contrast, for example: Ja mam zostać w domu, a on może wyjść.

Yes. Polish word order is more flexible than English word order. The version you have is neutral and natural, but other orders are possible for emphasis, for example: Lekarz powiedział, że mam zostać w domu i że moje ucho musi odpocząć.

What matters most is keeping the relationships clear: że should still introduce the subordinate clauses, and the sentence should remain easy to follow.