What is the ruling on looking at, shaking hands with, and playing with a female child who is not desired, given the possibility of her reaching puberty, and is it permissible to ask about her reaching puberty?

As long as the girl has reached an age where she is desired, she is to be treated as an adult. Thus, it is impermissible to touch her, look at her, or speak with her without a need. The jurists from the four madhhabs have explicitly stated this; a girl of nine years and above is desired by consensus, and an adolescent is like an adult in rulings.