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.
My husband told me that he concluded our marriage with a fake name that belongs to another person, because he was sentenced. Nowadays, he recieved an ID, passport and birth certificate with the his current name. What is the ruling on being married to him?
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. As regards being married to the person mentioned in your question: The marriage is valid so long as he was the same person your guardian concluded the marriage with, even if he changed his name i.e. if your marriage contract was concluded with the same person, since what counts regarding marriage is the persons not names. And Allah Knows Best.
When Iddah is Initiated for Deceased's Wife?
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.
Iddah is observed on the same date at which her husband passed away for four months, but as for the pregnant, it is observed at the date she gives birth. And Allah Knows Best.
What is the ruling on swearing a false oath by the Holy Quran?
Swearing a false oath by the Holy Quran dips the oath-taker in Hellfire, and one who had done so should turn to Allah in repentance, seek His forgiveness, give back rights to whom they belong, and pay the oath expiation.
