What is the ruling on buying a car from someone for less than the market price, because they are in a hurry to travel and want to sell their belongings at a lower price due to their urgency?
Yes it is permissible; however, a Muslim shouldn't take advantage of other's difficult situations.
Is it permissible for a woman to use a contraceptive without telling her husband although he wants to have as many children as possible and she can`t care for them?
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.
If the pregnancy is harmful to the woman, it is permissible for her to use a contraceptive; however, this has to be backed by a medical report from a God-fearing doctor. In addition, you had better inform your of what you are doing to avoid dispute. And Allah The Almighty Knows Best.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
Is it permissible to pay the Zakah to my grandmother who lives alone in a rented accommodation with bad conditions noting she had three sons.
Zakah cannot be given to ascendants (parents and grandparents). It is obligatory for her sons and grandsons to provide for her, and they would be sinful if they fail to do so. And Allah Knows Best.