Wipe is the more specific response. For example you can say "clean your room" but you wouldn't say "wipe your room".
No. To clean (청소- you can also use this term as a noun to say "chores") and to wipe (닦아) are different verbs.
Why not "brush"? as in brushing your teeth? Me: Brush your teeth kids. Kids: No! Me: Brush your teeth, please. Kids: No! Me: 닦아!