From your two examples I suppose Are you pure Italian would be correct but it doesn't sound quite right to me. Perhaps because, needless to say, the concept of racial "purity" is itself controversial... It's a question I'd never ask.
I agree, it would be better not to focus on one's race. The way to circumvent this restriction is to wonder whether their parents are both Italians by birth, whether the person you are talking to has been living in Italy for most of his life, etc. So generally your questions should not directly address the issues of purity. History tells us that contemplating such questions unfortunately leads to great disasters.