In question form, if DO is possible in the non-question form, then it MUST be used in the question. (You DO think he must have certain characterics, right? Well, then, what characteristics DO you think he must have?)
But, when a modal (can, may, etc) is being used , DO is never used, (question or not). (What characteristics must he have? Must he have ALL those characteristics?)
In other tenses (simple past, simple future), you'd use DID or WILL, also required in question form IF they're possible in declarative (non-question) form.
.... What characteristics did she think were required?
.... What questions will you ask?