yes and no... the batter can run if the ball hits the ground.. he is technically still out, but can get to and stay on first base.... most teams teach their catchers to touch the batter on strike three, regardless...... this puts the batter out no matter what
NOT on a foul tip.......It's a foul ball and all applicable rules regarding foul balls apply.......
A schwing and a miss on a third strike is an out even if it's in the dirt just as long as the catcher catches it.........but it's apparent in this case that the schwing was a foul tip.