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 a woman`s face an Awrah (a private part) that should be covered?

During prayer, a woman`s whole body is a private part except the hands and the face. Outside the time of prayer, scholars agree that a woman should put on her Islamic wear to avoid temptation.

A man has married a second wife and deprived the first from provision and overnight stay. He spent most of his time and wealth on his second wife. After sometime, the latter got sick and was diagnosed with breast cancer, which made him forsake her in bed and return to his first wife. What is the position of Sharia on this?

Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings involve acts of sin?

Maintaining kinship ties is mandatory, thus if visiting your blood-relatives hinders their sinful acts , then you should do so. However, if their sinful acts persist while you are at their gathering, then it is sufficient that you maintain kinship ties via telephone and the like.