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 swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
If a disposable water bottle has the logo (Al-Aziz Water Shop) ,is it allowed to throw it away considering that (Al-Aziz) is one of Allah`s beautiful names?
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.
You can remove some of the letters so that the word isn`t read as (Al-Aziz). And Allah The Almighty Knows Best.
Is it permissible to pay the Fitr Zakah (obligatory charity) of Ramadan on behalf of a dead person?
The Fitr Zakah of Ramadhaan isn`t due on one who had passed away before the sunset of the last day of Ramadan. And Allah Knows Best.