What is the question (do you think) inside the main question? Is there any name given to this sentence (do you think)? Is it correct to say "Whom do you think is he?"? If not, then, why?
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The correct form is "Who do you think he is?"
"Who is he" is not the main question. Grammatically, the main question is "Do you think ([that] he is who)?" ('who' is the unknown item in both questions). 'Do you think' has a question forming fronted auxiliary (from 'You do think'). The clause that is thought about ("he is who?") is secondary -- it does not change to the question format ("who is he"). The unknown item 'who[m]' gets fronted before everything. 'Whom' would be the accusative form, but 'who' is proper because it is a subject complement, and also acceptable because it has been fronted.