A woman used contraceptive pills, and her menses exceeded (15) days, is it permissible for her to fast?
The maximum of menstrual blood is (15) days, but if it exceeded that due to a medication, or the like, then it is Istihadah (bleeding outside the monthly period), and in this case she should perform Ghusl (ritual bath), pray, and fast. Moreover, she is obliged to make up for the missed fasting days which exceeded her regular menses. And Allah Knows Best.
Is it permissible to make up for missed fast after the beginning of the second half of the month of Sha`ban (the month before Ramadan)?
Yes, it is permissible, but one who had missed fasts should hasten to make up for them. As for the Hadith mentioned in this regard, the prohibition is for offering absolute voluntary fasting. And Allah Knows Best.
Ruling when Husband Utters Divorce Jokingly, or in the State of Intoxication or Anger
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The man who utters divorce while in the state of anger, his divorce doesn`t take place and the same ruling applies to the man who is forced to utter divorce because he is threatened to be killed or one of his organs cut off, in addition to other matters that scholars have mentioned in this regard. However, the man who utters divorce jokingly or while having intentionally drunk alcohol, his divorce takes place. And Allah The Almighty Knows Best.
Is it permissible for a Muslim woman to wear trousers at home?
Wearing trousers at home isn't prohibited so long as non-Mahrams do not see her. And Allah Knows Best.