Who do you think are? and What do you think are doing? are incomplete. The first "you" can't cover both parts in your sentences. Who do you think you are? and What do you think you're doing? are correct.
Compare:
Who do you think he is?
You certainly wouldn't be inclined to say: Who do you think is? in this context, would you? Yo
"Who do you think you are?" is correct. Its exact nuance depends on context, but often it is used accusingly or confrontationally. Similarly for "What do you think you're doing?"
"Who do you think are?" is unusual and normally wrong. It would need a special context in which something following "are" may be assumed, but even in that case "Who do you think is?" would be usual (e.g. "Nobody'