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 Making up Missed Obligatory Prayers an Obligation?
All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Missed obligatory prayer/prayers is a debt upon Muslim and making it up is an obligation, since the Prophet (PBUH) said: "pay the debt due to God, for it is the one which most deserves to be paid." [Agreed upon]. And Allah Knows Best.
Is it permissible for a woman who is observing `Iddah (waiting period) due to the death of her husband to travel for performing Umrah (Minor Hajj)?
A woman observing `Iddah of a revocable (Rajee`) divorce isn`t allowed to travel for Umrah except with the consent of her husband.
Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.