Almost. "The washing" is a noun that refers to the clothes that need to be, or just were, washed. "Do the washing" and "do the laundry“ are the same (ie. cleaning clothes). "Wash up" is a phrasal verb that refers to cleaning the dishes. I guess you could say "doing the washing up", but I think "washing up" is better.

Yes, doing the laundry means washing the clothes and doing the washing-up meaning washing the dishes. In the US we tend to say "doing the dishes" instead of "doing the washing-up".

"Doing the laundry" means "doing the washing". "Doing the dishes" means "doing the washing-up". One little word changes the sentence entirely. ;)