If the batter is still in the batters box, it is a foul ball, otherwise, the batter will be ruled out, and it is a dead ball with runners returning to their bases
This is wrong, if the ball hits the bat a second time in fair territory the batter is out, standing in the batters box means nothing in this rule. see MLB rule 6.05 h
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No.
The batter is indeed "out," but the play is not a "strike out" for either the batter or the pitcher.