Both are correct, the "got" version is probably more common in British English.

Both are correct and mean exactly the same thing. "I've got" is more common in British English (just like Chirstian says), and "I have" is a little more common in American English.

Either works IMO. I've got a sister is a little more casual. The word "got" and "get" in English function like super words. "I got up early this morning so I could get to work and get a lot done before it got too late to get some food from the store so I can get dinner on the table by the time my wife gets home so she doesn't get mad." LOL

It depends on the context. So 'I have a sister' is more of a declaration stating that you do in fact have a sister. While 'I've got a sister?' is infers that you did not know that you had a sister or implies a question.