Is housemaid right hand possession?
All perfect praise be to Allah, The Lord of The Worlds. May His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Housemaid isn`t right hand possession. Rather, she is a free woman because she is just hired to deliver a service. And Allah The Almighty Knows Best.
Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
What is the ruling on paying a sum of money to Saudi citizen in return of having Saudi residence? It is worth noting that this transation is labelled as a "worker's visa" but I am not going to work as such only to facilitate having it?
Allah The Almighty Says (What means): "O you who have believed, fear Allah and be with the truthful" [At-Tawbah/119]. False statements contradict truthfulness, and it is not permissible to write them. A Muslim has the right to live in any Islamic country, and those who prevent him will be accountable before Allah about their action and the justification for the prohibition. And Allah Knows Best.
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.