There is no single name for the process of answering one question with another or for the person who does so. However, here's a variety of possibilities for you to consider:-
counter-question(er), quasi-answer(er), verbal tennis(-player), query logic(ian), the Socratic approach (a Socrates). Socrates used questions to force his students to seek further questions and answers. Finally, there's aporia, which is a process in logic/rhetoric whereby a speaker reveals doubt...ie may be asking a question. Perhaps you'd care to pick one of these?
In the teaching circumstances you refer to, the Socratic approach is clearly the most specific, as already suggested above.