What should a woman who is in a state of menstruation, or confinement do if she became ritually pure during the day of Ramadhaan ?
If she became pure during the day time of Ramadhaan, it is desirable for her to fast the rest of that day, and to make up for it along with the days that she had missed due to menstruation.
What is the ruling on a person who broke the fast of one day in Ramadhaan because of hardship ?
He/she should make up for that day.
My husband told me that he concluded our marriage with a fake name that belongs to another person, because he was sentenced. Nowadays, he brought 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.
Is Iddah (waiting period) incumbent on an old lady whose husband had died?
Upon the death of the husband, Iddah is incumbent on the wife regardless of her age.