Does Husband fasting while being impure for having sexual intercourse with his wife at night permissible?
Yes it is permissible for you to fast because impurity isn't among valid fasting conditions, but you are obliged to perform ritual impurity in order to catch up Fajr prayer at its due time. And Allah Knows Best.
What is the ruling on a man leading his sisters in obligatory prayer, or Taraweeh prayer (Night prayer in Ramadan)?
The prayer of the man who leads his wife, or sisters in prayer is valid, but it is more rewarding for him to offer obligatory prayer in the mosque. And Allah Knows Best.
A man married a woman at the Islamic Centre in Brussels through a regular marriage contract. However, the husband left her for two years now and never provided her with financial support. Currently, she is staying in Amman, Jordan, and wants to remarry. Is her first marriage considered void and what should she do to remarry lawfully?
All perfect praise be to Allah, the Lord of the Worlds; may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
This issue is within the jurisdiction of the Islamic courts and they have the final say regarding the dissolution of the first marriage contract if there is valid ground for that. Therefore, her first marriage remains valid unless a court decision says otherwise. And Allah The Almighty Knows Best.
Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.