"What does he want us to be doing"?

Is it correct to say:

"What does he want us to be doing"?

It is correct--but as far as it being 'appropriate,' that is, fitting as far as context...for conversation and casual correspondence, it's fine.

If you were posing the same question in a more formal piece of writing, however, you might want to use more precise phrasing, such as:

'What does he desire that we do?'
or
'What would he like for us to do?'

It really doesn't matter-the meaning is clear in all cases--it just depends on the setting and intended audience.

In certain contexts "what does he want us to be doing?" would be precisely correct. Supposing you are organising a ceremony; you might ask, "While the President is giving his speech, what does he want us to be doing? Does he want us to be saluting? Standing to attention? Having a cigarette behind the podium?" The use of the tense here emphasises the contemporaneousness of the two actions.