Is it permissible to fast after the eighth or ninth day of the menstrual period that exceeds ten days, and is what remains considered istihadah (non-menstrual vaginal bleeding) for which one must fast, and is it permissible to take pills to prevent menstruation in order to complete fasting the entire month of Ramadan?

There is no need to doubt menstruation. If its days do not exceed 15 days, then the woman is considered menstruating and should not fast or pray. Upon seeing purity, whether by dryness or the white discharge, one must perform ghusl (ritual bath) and consider oneself pure. Any brownish or yellowish discharge before the sign of purity is considered menstruation. It is not permissible to fast or pray before purity. Missed fasts due to menstruation must be made up, and it is recommended to do so promptly and consecutively. If blood or brownish/yellowish discharge continues for more than 15 days, it is considered istihadah (non-menstrual vaginal bleeding) which does not prevent prayer or fasting. It is permissible to take medication to prevent menstruation, as long as it does not lead to harm.