"What characteristics you think he must have." Or "What characteristics do you think he must have."?

Answer 1

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?

Answer 2

If it's intended to be a complete sentence, the second is correct. The first is correct only as part of a longer sentence. "Please explain what characteristics you think he must have."

Answer 3

The first one is a clause–part of a sentence–but it's incomplete as written. The second is a question, so you used the wrong punctuation. Questions require a "?" at the end. Both are worded correctly, but YOU have to know the difference. And use the correct punctuation.