To me the second question is a little unclear. If you wanted to ask why someone rarely goes out, you could say, "Why do you never go out?" or "How come you don't go out much?"

Both forms are correct but the nuances are different! The first one simply asks why aren't you going out. The second one emphasizes the REASON for you NOT wanting to go out. Depending on context, it also hints that you are really stupid for not going out and that the asker is smarter than you are! The tone of the second form also sounds more formal.

Hi there! As an American English speaker, both are correct. "Do you not" would be seen as more formal and probably a little old. "Don't" is a contraction of "do not." In your speech, I would recommend using "don't," because it's much more common. So really, above, you're asking the same thing only one uses the contraction and the other does not (doesn't!). Hope this helps! -AmyV

You have the not in the right place. There is no difference between the two except the second one sounds as if you were trying to make yourself very clear.

Thank you all for your answers. They really help!!! I'll put in practice your teaching :)