I am aware that answering student questions with further, leading questions is sometimes dubbed “Socratic,” but I am asking more broadly about all occasions where someone asks a question and, instead of an answer, receives another question in response. Has a word ever been coined to name this phenomenon?
Is it not true that counter-question is a good match that describes the exact structure that you inquire about? Can you have a counter-question without an initial question?
Also, wouldn't you agree, though it might be obvious, that the second question (the answering question) is called a rhetorical question?
Out of numerous figures that are are related to this type of address, if I use for example interrogatio and question my own answer, preferably with more style than I employ, am I not actually confirming and reinforcing the answer that I have given?
Other figures are: erotema, anacoenosis, anthypophora, dianoea, aporia, epiplexis, exuscitatio, pysma and ratiocinatio and some of them cover exactly the meaning that you mention in comments: challenging the initial question.
In case that the second question is not a rhetorical question, but a real question that is raised by the first question then I would say you are simply investigating the subject in search for stasis (and the term counter-question still covers it).