It does sound natural and correct. (Especially if you put extra stress on the word “you” at the end while speaking it aloud). An alternative might be “You don’t ask people questions you wouldn’t want to be asked yourself.” Using the reflexive is a little bit clearer in regards to the restrictive clause.

Sure. You could also say: "Don't ask a question that you yourself would not want to answer." (There is no need to use the plural "questions".)