Latin is right. It's either did you get my book or have you got my book. The difference is that in sentence #1 you ask whether the person has received the book or not; in sentence #2 you ask if he has it.
Quite so. "Did you get my book?" at some given time in the past, "did you fetch it at the library/bookshop yesterday as I had asked you to" "Have you got my book?" here and now, is it in your bag?