Tomorrow he will be washing his car - Завтра он будет мыть свою машину. He will wash his car tomorrow - Завтра он помоет свою машину. No? Seems like the first sentence is Future continuous and the second is Future Simple
"Завтра он помоет свою машину" means "He will wash his car tomorrow (from beginning to end)", whereas "He will be washing his car tomorrow" can mean either "Завтра он помоет свою машину" or "Завтра он будет мыть свою машину" (He has the intention to wash his car tomorrow). Usually it means the former, but not necessarily.
Tomorrow he will clean his car won't work. I guess wash would mean the outside of car specifically, and clean could mean clean out, like the inside of the vehicle.
Just like мыть, washing involves using plenty of water whereas cleaning can be done, for instance, with a vacuum-cleaner. So, basically, you are right.
"she cleaned the floor" is accepted, but "he will clean his car" is not. Please, explain to me, why?
"Ему помоют машину" (literally, "For him they will wash the car"). You may also say, "Он помоет машину на автомойке", in which case, using the word автомойка implies that помоет doesn't mean he will do it himself.