Is it permissible for a menopausal woman who has blood discharges to fast?

If she reached the age of menopause, and her menses stopped, but later she saw blood discharges that lasted a complete day and night (24) hours, then it is menstrual blood. Still, if these blood discharges lasted less than a complete day and night (24) hours, then it is not menstrual blood, but it is bleeding outside the monthly period (Istihadah), and it does not prevent her from prayer and fast, but she has to make ablution for every prayer.

I made a vow "If the Israeli troops retreated out of Gaza strip I will buy a cake from a certain bakery" and it happened. Am I obliged to buy from the same bakery?

You are obliged to buy from the same bakery. 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.

If the imam does not leave time for reciting Surah Al-Fatihah, is the recitation of Al-Fatihah waived for the follower?