Mahrams

Q: If the wife passes away, can her mother live in the same house with the son in law as she has to take care of the children?

Q: Can you please advise me on the following matter: 

A girl was married and is now divorced from her husband. There was a child (female) from that marriage. The girl is now going to perform Nikah with someone else. 

What is the position/relationship of the child (who is now reaching puberty) to her stepfather? Is the stepfather her mahram or non-mahram?

Q: If a female who was born out of wedlock and raised by her maternal grandparents as their own, who is her Mahram? 

Does her uncles, her mother's brothers become her Mahram? Will her brother, they share the same mother but not father become her Mahram? 

And are her male cousins, sons of her mother's sister's and brother's become her Mahram?

Q: A girls father in law wants to shake her hand when they go to visit. It's a habit of his to shake his nieces hands and hug them. 

On Eid he shook her hand and kissed it. The mother in law said he's doing like how Dada used to do for the children. 

Is this allowed ?

Q: Is it permissible for the son in law to hug his mother in law if she is not old?

Q: Are my maternal and paternal grandparents' brothers (e.g. naani's brother) my mahaarim? Someone said that they are not.

Q: I would like to know if purdah is compulsory in front of an illegitimate son of my fathers?

Q: Is a woman's foster children (i.e. the children she breastfed) who are female mahram for her biological brother?

Q: Is purdah (hijab) to be observed between the father of the husband and and mother of the wife?

Q: Does the daughter in law have to observe purdah in the presence of the father-in-law after her husband has died? (Meaning the father-in-law was the father of her late husband)?