Stomach is an internal organ where most of the digestion takes place. Belly is really another word for abdomen. In conversational language people sometimes use belly instead of stomach, e.g. they may say "I have a sore belly" when they mean they have a sore stomach. I have heard mothers say to their children "Your belly's full" when they really mean "stomach"

"Stomach" and "belly" are used interchangeably. However "stomach" is more polite than "belly" which is used as slang. Example if some one is overweight, people might tease them and say that they have a " jelly belly".

Not really, but belly sounds a little immature and weird. I would only use it while reffering to a dog. [EDIT] Also, Jura is very right. And again, little kids say belly, or if you are being very informal or playing around with your friend. For example, if they're ticklish, you can say "I poke your belly! :D"