When can a runner advance?

[Zoochy]

There is a base runner and the batter fouls off a pitch. Can the umpire throw the ball to the pitcher? Or does the catcher have to throw the ball back to the pitcher?
Assuming the runner retaggs the base. When is it legal for the base runner to try to advance to the next base? Is the ball "in play' once the pitcher has the ball? What if the pitcher doesn't field the ball cleanly? If the ball is given to the catcher, can he try to pick off the runner?
I am not a baseball umpire, I am a fan of the game and my son asks me these type of questions.