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.
  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.
  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.
  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
  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
  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
  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.
  if this mother is a Muslim, then he inherits from her. For more benefit, please refer to Fatwa 85264. 
Allaah Knows best.