Is it permissible for a woman to visit the doctor while in her `Iddah (waiting period)?
It is permissible for her to do that during day time, but it is impermissible for her to leave her house at night except for a sound reason. And Allah Knows Best.
A young man who works for a conventional bank has proposed to me. Is it forbidden to marry him because he works there?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
Since the Messenger of Allah (PBUH) said: "When someone with whose religion and character you are satisfied asks your daughter in marriage, accede to his request. If you do not do so there will be temptation in the earth and extensive corruption."
Marrying such a person is permissible because he is a Muslim, and Allah may bless him with another job in which no usury/interest is involved. And Allah The Most Exalted Knows Best.
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.
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.