Are little leaguers supposed to be taught to run on the third strike regardless of whether or not the catcher misses the ball?

Answer 1

No. There is no 'uncaught third strike' rule in Little League. A base runner can advance if a third strike is not caught but the batter may not reach first base due to an uncaught third strike.
Majors and minors can't, but juniors and seniors can.

Answer 2

Only if the catcher doesn't catch it in the air.