Assalamu Alaikum ,
Please advise your Fatwa on the following :
1. Does an Heir (woman) married to a Christain man ( Civil Marriage) living in a Christain Soceity is entitled to inheritance from her mother . Is her marriage Valid ?
Status of Her Children ?
2. Does an Heir (man) NOT performing SALAH , Even Fridays will be entitled for Inheritance from his mother .
Jazakum Allah Kheiran
Haj Abu Saleh

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

Your question included three matters:

1-    The marriage of a Muslim woman with a Christian man: This marriage is absolutely void because the scholars agreed in a consensus that the marriage of a non-Muslim man with a Muslim woman is void. For more benefit, please refer to Fatwa 83101.

2-    The inheritance of this woman from her mother: If her mother is a Muslim, then she inherits her, but if her mother is not Muslim, then she does not inherit her. The Prophet said: "A Muslim does not inherit a non-Muslim and a non-Muslim does not inherit a Muslim." [Al-Bukhari and Muslim] As regards the fact that that woman lives in a Christian society or that she is married to a Christian as you claimed in the question, then this does not affect the matter of inheritance. However, she is considered as a fornicator because of (entering into) this marriage.

3-    The right of a person who does not pray in the inheritance of his mother: The scholars differed in opinion about the ruling on a person who abandons the prayer. The view of the majority of the scholars is that he does not go out of the fold of Islam, however, some scholars are of the view that he goes out of the fold of Islam. Based on the view of the majority of the scholars if this mother is a Muslim, then he inherits from her. For more benefit, please refer to Fatwa 85264.

Allaah Knows best.