Is it permissible for the doctor to uncover the hair of a female patient out of necessity?
It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams (unmarriageable kin).
Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
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 person who had vowed to give a certain amount of money to another, but didn`t find the latter to donate that money to the mosque?
In principle, the vowing person should abide by his vow as much as possible. Therefore, if he couldn`t find the person that he had made the vow for, then the vow itself is countless and nothing is due on its maker. However, if the latter happens to find the former later on then, he has to give him that money.