What happens when the pitcher catches a ball from the batter?

Answer 1

As soon as the bat hits the glove, the ball is dead. The batter is awarded first base, no matter if s/he hit the ball or not.

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Above is incorrect. The play is not necessarily immediately dead on catcher's interference. If the batter puts the ball into play, the umpire lets the play continue until all action has stopped before calling time. The Manager of the team at bat then has a choice of either taking the result of the play, or having the catcher's interference enforced, which would put the batter on 1B, and all runners forced to advance would advance. However, if the batter reaches base safely, and all other runners advance at least one base, then the interference is ignored.

Answer 2

The player must have control of the ball in the glove. If the glove comes off before he completes the catch, it is not an out. The glove is considered part of the fielder's hand and he must have his hand in the glove.

Answer 3

the batter is out