G.B.
All cats poop. Most cats use a litter box. Some cats use the toilet. So yes, cats can be trained.
Buy a spray bottle that has a straight shot. You know, one that shoots a straight single stream all the way across the room.
When the cat starts to do something you don't like you secretly shoot the water at them. They don't see it coming from you, they start associating water, getting wet, not liking that area.
As for cats on the counter, cats are among the cleanest animals. If you just wipe the counter off before you start a task with some sort of kitchen cleaner it's clean.