Would "che vuole?" or "cosa vuole?" also be acceptable ways to translate "what does he want?"
yes, all of them are correct, but the most correct one is "che cosa vuole", the others are a bit less formal/less polite/more colloquial.
My answer was "what do you want" which I thought, as soon as I hit the button, would be marked wrong. It wasn't. Why?
I did the same thing. In hindsight it should be wrong. “Che cosa vuoi?" would be what do you want.
Fast forward one year and still, no. Literally translated we're correct but I guess our moderators think that's being too literal. I'll flag it again.
why is this 'what does HE want' - does that mean it could also be 'she'? I put 'what do you want' which was accepted...but the gender thing still confuses me...qualche volta!
Yes. For lui/lei/Lei, volere conjugates to vuole. It accepted "what do you want" because "Lei" is the formal you. As you said, it would also accept "what does he want" and "what does she want".