Learn Polish: Mimo że jestem zmęczony, dbam o siebie. - Even though I am tired, I take care of myself.

Because it comes directly after a preposition (o). The short form się can never be used after a preposition, so we must use the full, independent form siebie instead.

Which case is siebie taking in this sentence?

It is in the accusative case. The verb dbać (to take care) always pairs with the preposition o followed by the accusative. Conveniently, the accusative form of the reflexive pronoun is simply siebie.

Could I use chociaż instead of mimo że?

Yes, absolutely! Mimo że and chociaż both mean "even though" or "although" and are completely interchangeable here. You might also come across pomimo że, which means the same thing.

How would a woman say this sentence?

A woman would say Mimo że jestem zmęczona, dbam o siebie. The adjective zmęczony (tired) changes to the feminine zmęczona, but the pronoun siebie stays exactly the same because it applies to all genders and persons.