Zero outs, runner on third. Batter hit sharp line drive at pitcher. The batted ball hits the pitcher's glove and deflects high into the air and away from the pitcher. Anticipating the pitcher might catch the line drive, the runner on third retreats back toward third. Upon seeing the ball deflect off the pitcher's glove, the runner on third, retouches third and immediately advances and scores. As the runner is advancing, the second baseman catches the deflected ball before it hits the ground and throws the ball to third baseman, who promptly tags the third base. Is the batter out? Is the runner out?

Answer: The batter is out. The runner is not out and therefore his run counts.

Under Official Baseball Rules the relevant rules concerning the batter are:
6.05 A batter is out when— (a) His fair or foul fly ball (other than a foul tip) is legally caught by a fielder.
2.00 A 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....
2.00 IN FLIGHT describes a batted, thrown, or pitched ball which has not yet touched the ground or some object other than a fielder.
Therefore, the batter is out because a fielder has caught the ball in flight.

The relevant rules concerning the runner are:
7.08 Any runner is out when—(d) He fails to retouch his base after a fair or foul ball is legally caught before he, or his base, is tagged by a fielder.
2.00 (Catch) Comment: A catch is legal if the ball is finally held by any fielder, even though juggled, or held by another fielder before it touches the ground. Runners may leave their bases the instant the first fielder touches the ball.
Therefore, the runner is not out when the third baseman tags third because the runner retouched his base after the pitcher touched the ball. The runner does not need to retouch again after the ball is eventually caught.