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 for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.
Is it true that if a ritually impure, or a menstruating woman visits one who is in confinement, harm will be inflicted on the latter, or her baby?
This is completely wrong and unfounded piece of information.
There is a car dealership, and their method of selling is as follows: The customer pays the full price of the car, the car is registered in their name, and another contract is written in which the company commits to refunding the full price of the car to the customer after one year from the purchase date. What is the Islamic ruling on this transaction?
As for purchasing the car, paying its price, and owning it, this is a valid sale with no issue. However, the condition of refunding the price — whether the car is returned to the seller or not — is an invalid condition that nullifies the entire contract. Therefore, this transaction is invalid and not permissible. And Allah Knows Best.