I did that during a neighborhood pick-up game when I was a kid, but we were using a really old ball and the stitching was already half gone. Hey, it was the only ball we had! But I've never heard of it happening in an organized game, MLB down to Tee-Ball.
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Yes, this has happened enough to have a specific rule about it. If the bat strikes the catcher or his equipment (almost always his glove) while the batter is swinging, the umpire calls catcher interference and the batter takes first base.
Yes, and it happens once or twice a year. Paul Konerko was hit in the face just last week on Sept 16th 2010.