Why is "Have you already done the dishes" wrong?
We require "those dishes", see my other comment.
Or, have you washed those dishes yet
Here we require "those dishes" for "to nádobí". As "to wash the dishes" is just "mýt nádobí".
Wash/Do the dishes! Umyj nádobí!
Why is it not “...ty nádobí”?
It's a collective noun and it's singular neuter.