Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
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).
What is the ruling on not offering prayer in a nearby mosque, and going to a distant one?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Praying in the nearby mosque takes precedence unless there was a circle of knowledge, or a larger congregation in the distant mosque provided that the congregational prayer in the nearby mosque isn`t undermined. And Allah Knows Best.
What is the ruling of Sharia on male doctor delivering a baby?
All perfect praise be to Allah the Lord of The Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The person most entitled to deliver a baby is a Muslim woman, then a woman from the People of the Book, then a woman from the disbelievers, then a Muslim man, then a man from the People of the Book. However, if there was no woman to deliver the baby, then it is permissible for a male doctor do deliver it. And Allah The Almighty Knows Best.