Training your cat to use a human toilet is an involved process that can take a while. I have seen the litterboxes that you put on top of the toilet at the start. I don't have any personal experience doing this, so I'll leave it to others to tell you the steps involved.
I know cats can definitely flush. Ginza does this from the floor and pressing down with his front paws too like BJohnson's sister's, or the preferred method is to jump up onto the water reservoir and step down on the lever with one paw. It definitely startles me when I think I'm the only one in the house.
EDIT: I just wanted to clarify that 1. I don't train my cats to use the toilet, they use litterboxes. 2. Ginza flushes for fun to amuse himself.