Is it permissible to give lessons to mixed students of the seventh grade?

It is impermissible to give private lessons to mixed students since such an act leads to serious consequences for the students in this grade are teenagers.

What is considered a woman`s Awrah (parts of the body that must be concealed during prayer) while offering prayer, and is her foot part of it?

A woman`s whole body is considered an Awrah while offering prayer except the hands and the face. Thus, her foot is an Awrah as well. And Allah Knows Best.

Is it permissible for the guardian, or the father of the child to hit the latter in order to teach him/her good manners?

It is permissible for the guardian to hit his/her child for teaching them good manners once they become ten years old provided that it is done mildly, harmlessly and without intimidation. Actually, such punishment is to be employed gradually i.e. after: asking, warning, and scolding. As for the teacher, he isn`t allowed to hit the student unless the latter`s father approves of it, and it is done according to necessity without intimidation, or harm.

Is it allowed to shave my beard to join the army taking into consideration that I can`t find a job?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

The army should be joined in preparation for Jihad (Fighting in the cause of Allah), so if Jihad contradicts with growing a beard then it is allowed to shave it. And Allah The Almighty Knows Best.