The answer to our first question if pretty simple: One experiment.

Theoretically, you would only need one apple going up to the sky when you throw it out from your hand to disprove the theory of gravity. This would oblige scientist to redefine the theory of gravity to exclude the specific case where your apple goes up into the sky. For this a theory needs to be repeatable and falsifiable. If I would tell you that there's a really small teapot between the Earth and the Sun and so small we can't detect it, how could you prove me wrong? However, if I'd tell you that all birds are white. You could try to prove me wrong, look  and show me that there's a bird out there that is black, for example, a crow.

Regarding proving a theory, it's more hard. You can't never be sure that you are not missing a case where your theory doesn't hold up. The best you can do is to solidify your theory trying to cover the biggest amount of cases.