The batter is only out on a dropped third strike when there are less than two outs and first base was occupied at the start of the pitch. With two outs the batter may take first base on a dropped third strike even if it was occupied at the start of the pitch.
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The batter or the runner? Either would have to be tagged to result in an out. The catcher (or any player) would have to have the ball and touch the runner from third. If you mean the batter is still in the batter's box and the ball rebounds and hits him then I would surmise that it is a judgment call by the umpire as to whether or not the batter was in the way of the catcher being able to make a play.
Yes. The game is in play unless someone call time out. Any runner can still advance although if forced, the runner on first or the runners on first and second get free passes. The batter can take his time going to first but all other runners are in play. That's why you never see a catcher trying to catch a the runner going from 1st to second on a steal because on ball 4, if there's an error on the throw, everyone can still advance.
Yes. If the batter hasn't been struck out or advanced in anyway, the same batter gets to come back to bat at the beginning of the next batting cycle.