"Do the washing-up" should be accepted. It is perfectly good English
No, no, no. First, your ear is not trained enough and you just heard "повсюду" instead of "посуду" but there is no "в" here. Although I think "повсюду" sounds like "посуду" for a foreigner. "Посуду" is translated as "dishes" and "повсюду" - as "everywhere". It would be very funny: I asked you to wash the dishes and when I'd come back I saw you washed everything. Wow! Cool! You're badass! // Second, "помои" means "slops" and "помой" means "wash". And your sentence "Дима, помои повсюду" is translated as "Dima, the slops are everywhere". If you were joking, that was a great joke, thank you))
- You can clean the table, but you can't clean the dishes, you can only wash it. If there some grease, dirt, peaces of meal, rotten tomatoes on the dish - in this case, yes, you can clean it with napkin or bare hands, but after all, eventually, you have to wash it.
- Tableware, I think, is very general and official name for dishes, like in some old prim estate somewhere in England, where valet addressing to his master say: "Sir, use the tableware in a proper maner, please". Concept of "dishes" in ordinary life is in use more often. Anyway - it is just my thought.