I don’t own my house. I rent (it). I don’t own a house. I live in a rental. I don’t have a house. I have an apartment.

1 & 2 are both correct. 3 is interesting: it can suggest that you have accepted the fact that you must rent. "I live with" is often used to mean "I accept"

1 and 2 are correct. number 3 is wrong, you life with somebody else, your friend, room mate, another person who shares the rent, but not with the rent

Numbers 1 and 2 are correct.