Regarding "the dishes":
A dish, in my area of the USA is more often called "a plate". It is a flat, usually round thing made from plastic, glass, or ceramic that you put non-liquid food on and eat from.
However, the dishes can either mean a collection of plates, (especially as compared to bowls, as in "put the dishes here and the bowls there"), but most often means all the things that are used for eating and drinking in the phrase, "do the dishes". To do the dishes means to clean all the plates, bowls, cups, glasses, forks, knives, spoons, pots, and pans or rinse them and put them in the automatic dishwasher to be cleaned. I believe in British English this is often called "washing up".
For example, in the song about Cinderella from the Disney animated movie Cinderella, they sing about her doing, "the dishes and the mopping/they always keep her hopping". In that case, the dishes means all the tableware as I described in at the end of the previous paragraph.
"Silverware" and "cutlery" are commonly used in my area of the USA to mean forks, knives, and spoon that are used at the table for eating. "Flatware" means the same thing (in my experience) but is rarely used in a normal household. "Tableware" is most often used in stores, catalogs, and websites that sell kitchen and dining implements to indicate all the things that are used to eat at the table as opposed to things used in the kitchen to cook food. "Flatware" is also used in the same contexts to mean the forks, knives, and spoons used at the table for eating, as opposed to any forks, knives, or spoons used for cooking, which are usually larger (and sharper, in the case of knives).
I rarely hear "cutlery" used in homes, but sometimes in restaurant the sets of silverware wrapped up in napkins that are placed at the tables are called "cutlery" or "cutlery sets" by staff or customers.
To summarize my answers to your two questions:
There is no word I use for all of those things that doesn’t also include things that don’t usually go on the table. But if you include the pots and pans and so on that are in the kitchen, then it’s “the dishes”. How do you get everything off the table to “do the dishes”? You “clear the table”. When getting ready for a meal, you “set the table”. What do you set the table with? “Silverware, plates (and/or bowls), and glasses”. There is no one word I know of that is used commonly to mean all the things that go on the table.