Only if it is on a third strike. That would be considered a foul tip and, by official rule, is different from a foul ball. A foul tip is when the catcher catches a ball that is 'tipped' by the batter and the ball goes straight back into the catcher's mitt. If the catcher drops the ball it is considered a foul ball and not a foul tip.
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No, The ball must be caught "in flight". According to Official Baseball Rules: Rule 2.0 Definition of TermsA CATCH is the act of a fielder in getting secure possession in his hand or glove of a ball in flight and firmly holding it; providing he does not use his cap, protector, pocket or any other part of his uniform in getting possession.IN FLIGHT describes a batted, thrown, or pitched ball which has not yet touched the ground or some object other than a fielder.Rule 10.05 Base Hits(a) The official scorer shall credit a batter with a base hit when:(1) the batter reaches first base (or any succeeding base) safely on a fair ball that settles on the ground, that touches a fence before being touched by a fielder orthat clears a fence;All bold emphasis is mine.
Rules 6.08(d) and 7.08(f) say that if a fair batted ball hits a runner in fair territory before touching or passing an infielder, then the runner is out for interference. The ball is dead, and all runners must return to the base last occupied, unless forced by the batter who is awarded 1B. As soon as the ball hits the runner, it is dead, so it doesn't matter what happens...caught or not caught...after that.
no, but the base runner is out if he is struck by a BATTED ball (but he isn't out if he touches a ball thrown by a fielder)
Depends. If the ball bounces of a defensive player, and is caught, the batter is out. If it hits a runner, the ball is dead and the runner is out but the hitter is awarded first base.