I do ask my T a lot of questions- about his family/childhood/personal/professional life- well anything. Sometimes he answers sometimes he doesn't. Depends more on my reason for asking the ? than on the ? itself.
I would not be with a T with blank slate approach (not sure if it's the correct term but I know there is a therapeutic school build on that). I'd feel inferior, couldn't take my T as a real person and in the long run it would only push me to ask more (or surf the net for answers)

Still I do understand there are some ?s my T doesn't want to answer, but do you have kids is not one of them. BTW my T doesn't talk about himself ever, I have do the asking and even then he is able to transfer the focus back on me. He is very good at changing the subject, better than I am- surprisingly, cause I have considered myself to be an expert in this area.