Should vowed fast be offered in consecutive days, or not?
Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.
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.
Is my husband entitled to take my salary?
Your salary is yours, and you may give some of it to him as a kind of free-will contribution.
Is it permissible for a person to sever ties with his sister because her husband had quit prayer and done him harm?
A sister is a blood-relative and maintaining blood-relations is obligatory in Islam. As regards her husband, he should be invited to the way of Allah with wisdom and beautiful preaching.