What is the ruling on women playing sports and teaching physical education classes at girl schools?

This is generally permissible, as long as there is no intermingling between the two sexes and the girls are alone at the school. Exercising is important to build the body, protect the mind, and enhance performance. It is something desirable, unless there is a Shar`i impediment, such as a man and a woman being in privacy or requiring a woman to take her clothes off.

I would like to advise Muslims against overstrict observance of the rule of blocking possible ways to evil. It is favorable to keep away from imposing unjustifiable constraints over people in something that may be involving no evil at all. If such rule is not applied for some interest, such interest must be significant and reasonable.