Yes, it's correct. It's similar to saying "Has Tim left the party yet?". With "yet", you are/were waiting for him. With "already", he's gone earlier than you expected. For example: Have you done the coursework already? Why did you do it so early? The deadline isn't until next month. Have you done the coursework yet? You had better hurry up. It is due tomorrow.

Yes it is correct, the word already is used in interrogative sentences as well! :)

You can use "already" in questions, but it implies that you'll be surprised if the answer is "yes." The use of "yet" makes for a more neutral question.