I'm not surprised that I got this one wrong! It seems like a weird way to ask where someone works, but maybe it goes back to a time when most people who worked were actually "making" something... Anyway, could one also say, "Kde František pracuje"?
Kde František pracuje? is perfectly fine. :)
"Kde František pracuje?" je to samé co "Kde František dělá?".
"Kde dělá" is coloquial, but it's the same...