Gatsby is probably asked this question routinely and most likely by strangers. He is not used to having a friend, and he immediately apologizes after that automatic reply shoots out of his mouth. Nick was making casual conversation, but Gatsby is unaccustomed to those routines since he has withdrawn from society (he never mingles at his parties, either).

Because one of the themes of Gatsby's life is this attempt to break into the posh set and of course in the end to work things out so that Daisy runs off with him, he is incredibly concerned about his image.  He has worked so hard to cultivate this very cultured image and he could ruin that by allowing others to know how he got his money.

It is absolutely unacceptable among Daisy's crowd to have had to work for your money, especially if it is tied to anything improper or illegal.  It is perfectly acceptable to have huge parties with alcohol, etc., but to be connected to smuggling it would be a mark on his character that nothing could undo.

As the previous post says, definitely rude, but Gatsby can ill afford to tell the truth.  Of course we know that he also struggles to lie to Nick (as everyone does according to Nick) so he also can't just make something up, so the response is a rude one instead.

This happens about half way through Chapter 5.  Gatsby is talking to Nick while Daisy is off "washing her face."  Gatsby is about to show Daisy his home.

I think that Gatsby answers in this way because he made his money illegally.  He seems to have been involved in smuggling alcohol with people like Meyer Wolsheim.  So he does not want Nick to talk about that, especially when Daisy might hear.

This is not an appropriate reply because it's pretty seriously rude to tell someone to mind their own business when they ask what you do for a living.