1. Is my sister entitled to a share of what my father left behind?
2. Is there anything wrong with her marrying a Christian man and saying that her children are free to choose between Islam and Christianity?.
The factors that determine whether one person inherits from another are three: blood ties (kinship), marriage and wala’ (relationship between a freed slave and his former master). And the impediments to inheritance are three: being a slave, being the murderer of the deceased, and being of different religions. The fact that your sister ran away from the family home does not alter the fact that she is the daughter of your parents. This means that she comes under the heading of kinship by virtue of which she is entitled to inherit from them so long as she has not apostatised from Islam.
The fact that she ran away from the family home does not deprive her of her right to inherit from your father, even if she has stayed away from her parents’ home for a long time. And we do not know of any difference of scholarly opinion concerning that.
Marriage of a Muslim woman to a non-Muslim man is a major sin, and there is scholarly consensus that it is haraam and that the marriage contract is invalid. Allah, may He be exalted, says (interpretation of the meaning):
“And give not (your daughters) in marriage to Al-Mushrikoon till they believe (in Allah Alone) and verily, a believing slave is better than a (free) Mushrik (idolater, etc.), even though he pleases you”
“O you who believe! When believing women come to you as emigrants, examine them, Allah knows best as to their Faith, then if you ascertain that they are true believers, send them not back to the disbelievers, they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked: A man married a Muslim woman then it became apparent that the man was a kaafir. What is the ruling on that?
If it is proven that the man mentioned was a kaafir at the time of the marriage contract and the woman was Muslim, then the marriage contract is invalid because it is not permissible, according to the consensus of the Muslims, for a kaafir to marry a Muslim woman, because Allah, may He be glorified, says (interpretation of the meaning): “And give not (your daughters) in marriage to Al-Mushrikoon”[al-Baqarah 2:221] and “then if you ascertain that they are true believers, send them not back to the disbelievers, they are not lawful (wives) for the disbelievers nor are the disbelievers lawful (husbands) for them”
End quote from Fataawa Islamiyyah, 3/230
It says in a statement of the Islamic Fiqh Council: Marriage of a kaafir to a Muslim woman is haraam and is not permissible according to scholarly consensus. And there is no doubt concerning that, because of what is implied by shar‘i texts.
End quote from Fataawa Islamiyyah, 3/231
Any Muslim woman who is aware of that (and still does it) deserves to incur sin and deserves to be subjected to the hadd punishment for zina. But if a woman was unaware of this ruling, sin is waived in her case, but it is not permissible for her to remain in that marriage, because it was never valid in the first place.
See also the answer to question no. 8396
It seems from your question that your sister has not declared any apostasy from Islam, but there remains the issue of her marriage to a non-Muslim and her saying that her children are free to choose between Islam and Christianity. So we say:
1. It seems that her marriage to a Christian man is an invalid, haraam marriage. If she did that regarding it as permissible when she was aware of the prohibition on it, then that constitutes apostasy from Islam, and the apostate cannot inherit from his Muslim relative. If she did not regard it as permissible, then she is a sinner, as mentioned above, but that does not prevent her from inheriting.
2. With regard to her saying that her children are free to choose between Islam and Christianity, that is a false notion and a great evil, because Islam is the true religion and nothing other than it is valid or acceptable from anyone.
It should be noted that one of the things that invalidate one’s being a Muslim and that constitutes apostasy is not regarding as disbelievers the People of the Book and the mushrikeen, or having some doubt as to whether they are disbelievers, or regarding their religion as valid. See the answer to question no. 31807 and 6688
What you have to do is advise your sister and explain the seriousness of what she has said, and call her to leave her husband if he does not become Muslim.
To sum up the issue of her inheritance: if she remained Muslim and did not regard as permissible her marriage that she knew was invalid, and she did not regard the religion of the Christians as valid or doubt that they are disbelievers, then she may inherit. Otherwise if she has done any of these things that invalidate her being a Muslim, then you have to advise her and ask her to repent in front of scholars in your country. If she repents, all well and good, otherwise she is an apostate and has no right to inherit from your father.
And Allah knows best.