Ruling on performing acts of righteousness for a disbelieving father after his death

This message reached us from Iraq, from the Iraqi Republic, Wasit Governorate, from the Muslim Hadi Muthashir. He says in this message that he is asking about being dutiful to a disbelieving father after his death. He says: My brother and I have believed in Allah, the One, the Unique, and to Him be praise that He guided us. We live in a village filled with disbelief, shirk (associating partners with Allah), and injustice, especially the family in which my brother and I live. And I present to you my problem, which is:
Our father passed away while we were children, and we had a mother who did not believe in Allah the Exalted, and who was full of disbelief. When we asked her about our father, whether he was a believer or not, she told us: He used to steal people’s things and was not a believer in Allah. Rather, he would say with his tongue: “Allah is our Lord and Muhammad is our Prophet,” but he did not establish the prayer. Frankly, he was, as people say, and especially our mother: he did not believe in the Last Day. And when he died, he left neither money nor lawful wealth for me or my brother. My question is:
First: Should I slaughter sacrifices for him, or is that forbidden for me?
Answer: If the mother is not lying in this matter, then such a person is not to be sacrificed for, nor is forgiveness sought for him; because he left off the prayer, and one who leaves off the prayer is a disbeliever according to the most correct of the scholars’ opinions. However, if there is someone else who knows him, and you ask about him, and you find someone who knows that he used to pray and that he was upright, then there is no harm in sacrificing for him, no harm in making du'a (supplication) for him, seeking forgiveness for him, and giving charity on his behalf. As for her, she is not trustworthy as long as she is a disbeliever who does not believe in Allah and the Last Day; she is not trustworthy in reporting news. But if there is something that corroborates her, in the form of circumstances and evidence that support what she said, or the testimony of some people, then he is not to be sacrificed for, and it is not affirmed that he was a disbeliever. However, if there is nothing here except the mother’s testimony, and there is nothing to support it, and it is known that he testified that there is no god but Allah and that Muhammad is the Messenger of Allah, and that he mostly displayed Islam, then it is not permissible to leave off giving charity on his behalf and making du'a for him. Rather, du'a should be made for him and charity given on his behalf, until it is known that he was engaged in something of the things that make one a disbeliever, which take him out of Islam; either leaving off the prayer, or mocking the religion, or insulting the religion, or something similar to that from the nullifiers of Islam.
In that case, no du'a is made for him, no prayer is offered for him, no charity is given on his behalf, and no sacrifice is made for him. As for her alone, it is not sufficient; because the testimony of a disbeliever is not considered. But if there is something to support her, in the form of circumstances or the testimony of others, that he did not pray, then no sacrifice is made for him, no charity is given on his behalf, and no forgiveness is sought for him.
The presenter: The sender, Hadi, says: But he did not establish the prayer. Frankly, he was, as people say, and especially our mother. People say to him: Your father...?
The Shaykh: If it is well-known among the people that he did not pray, then no du'a is made for him and no sacrifice is made for him. Yes.