"I do not ask what he does."
Translation:Non chiedo cosa fa.
No, it doesn't sound like Italian, sorry. :(
"Non chiedo che cosa lui fa" "Non chiedo cosa lui fa" could be acceptable, but they both sound pretty awkward.
"Non chiedo che fa" that sounds fine.
"Non chiedo che fa lui" "Non chiedo che cosa fa lui" "Non chiedo cosa fa lui" are also acceptable.
I now check if there is something online to explain it. But I cannot find anything easily.
Here it is! About saved me!!!
In questions beginning with an interrogative word, the subject is usually placed at the end of the sentence. Quando guarda la TV Michele? - When does Michael watch TV?
(The given sentence is not a question, but since it starts with What, that in as interrogative pronoun, it acts in the same way.)