What is the ruling on a woman driving a car?

All Perfect Praise is to Allah, The Lord of The Worlds, and may His Peace and Blessinsg be upn our Prophet Muhammad and upon all of his family and compnions.

It is permissible for the woman to drive a car, but it is impermissible for her to travel unaccompanied by a Mahram (Unmarriageable kin). And Allah The Exalted 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 the one who doesn`t offer prostration of recitation during Quranic recital considered sinful?

Prostration of Quranic recital is a Sunnah due on whosoever recites a verse that contains a prostration. It is also a Sunnah for the reader and the listener. However, none of them is considered sinful for not offering it, but they deprive themselves from a great reward. It is obligatory to offer the prostration of Quranic recital in congregational prayer following the lead of the Imam.

What should a worshiper who has forgotten a pillar of prayer do?

If he/she remembered the forgotten pillar before offering it in the following Rakah (unit of prayer), then he/she should offer it, and continue the prayer, then offer Sujood As-Sahw (Prostration of forgetfulness) at the end of it. But, if he/she remembered that forgotten pillar after having offered it in the following Rakah, then he/she should offer a new Rakah in its (the one in which he forgot a pillar) place, and offer Sujood As-Sahw.