Although what do you care and why do you care have similar meanings, they have almost opposite uses.
Why do you care? is employed as a real question: someone is concerned about something which seems innocent or trivial to you, and you ask for a reason. It is ordinarily spoken with emphasis on why and care (but see snailboat's comment, below).
What do you care? is employed dismissively: it implies that the concern 'you' express is not something which touches 'you', and therefore 'you' should not meddle in the matter. It is spoken with emphasis on you.
Ahmad, it follows the same rules as the past question.
All tenses mentioned can be used in that sentence. However, as it's a past tense and you are telling a past history, using "had" would make it sound more fluent.
I waited until he had left the room, then I entered there.
Now talking about your second question.
I am waiting until he comes back - Right
I am waiting until he will come back - Wrong
I wait until he comes back - Right
Remember, "until" cannot be used followed by the future tense, all the others alternatives are suitable for until, past tense, past continuous, present tense, present continuous, present perfect..
And I waited for him to leave the room also works and sounds good too, and functions with others tenses too:
I was waiting for him to leave the room
I will be waiting for him to leave the room;
I waited for him to leave the room
I have been waiting for him to leave the room.
Best Answer
The phrase the night before last night is exactly how I'd reference it, although, in many contexts – and that includes informal contexts – I'd typically leave off the second night: