(a) is correct English, but is markedly less usual and needs some kind of prior context. E.g.:
— I spend all my time writing. — What do you write? (i.e. "What kind of thing do you write?")
(d), on the other hand, can be asked without any prior context, when someone is observed to be writing. If you used (a) in that situation then you would sound like something out of a pe