What is the ruling on not fulfilling a vow?
A vow is a commitment towards Allah, The Almighty, and so, it should be observed as much as possible, and it is impermissible to take it lightly. Allah, The Exalted, Says –in commending the believers (What means): "(Because) they perform the vow and fear a day whereof the evil is wide-spreading." [Al-Insan/7].
A woman went for Hajj while pregnant, gave birth after leaving Arafat, and was unable to perform the Tawaf Al-Ifadah. If she cannot stay until she becomes pure from postpartum bleeding (Nifas) and she is from a distant country, what is she liable for?
This woman is advised to leave Mecca in a state of Ihram if the caravan departs. Once she reaches a distance where returning to Mecca is not feasible, she should consider herself in a state of being prevented (Muhsar). She may then exit Ihram, and she is required to offer a sacrificial animal (Blood) at that location. Additionally, she must perform a compensatory Hajj at the earliest possible opportunity. And Allah Almighty Knows Best.
Is it permissible for a person who broke fasting in Ramadan because of being sick to pay a ransom?
If there is hope for him/her to be cured, then he would be required to make up for the missed fasting days when he/she recovers, and no ransom is due on him/her. If his/her disease is incurable, then he/she would be required to pay a ransom for each missed fasting day, which is (600) grams of wheat, or rice. 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.