Depends on the species, basically the largest rodent or food item that they can eat, whatever is up to just a little bigger around than the snake is at its thickest point. Normally a pinky or fuzzy mouse. There are some species of very small colubrids that you have to feed mouse heads or lizard tails for a bit.
Rarely. It really does depend on the species. Most snakes are attracted to live prey naturally. It can be somewhat tricky to get a snake to take whole pre killed prey sometimes, much less a "piece" of something. It also depends on what scent attracts the animal. Many snakes feed on lizards and small reptiles as babies in the wild, some on chicks, some on fish, so with some species a tactic called "scenting" is used, rubbing a pinky mouse with chicken, fish, lizard, or the blood of the animal, or actually just dipping it in chicken broth, can entice them to eat as well.
Many years ago, I et a guy who fed his two adult Burmese pythons quartered chicken pieces from the grocery store. That was the damnedest thing I've ever seen. I'm still not sure it was the healthiest idea, but it was weird to watch.