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.
The person whom you mentioned in the question does not inherit his mother if she is a non-Muslim, and she does not inherit him as well, because a Muslim does not inherit a non-Muslim and vice-versa. However, if she is a Muslim, then she inherits him and he inherits her.
As regards family lineage, then he is not traced back to the fornicator or adulterer, rather, he is traced back to his mother if she was not married when she committed Zina (fornication or adultery), and he should be traced to the husband of his mother if she was married when she committed Zina. In any case, he is not traced back to the fornicator (or adulterer) whether or not his mother was married at the time of Zina, and he does not inherit him as we clarified in Fatwa 89082.
However, some jurists are of the view that if an unmarried woman commits fornication and gives birth from Zina and the man who committed Zina with her asked that the child be traced back to him, then he is traced back to him; this is the view chosen by Ibn Taymiyyah and Ibn Al-Qayyim .
Furthermore, Imaam Abu Haneefah said: ‘‘I do not see any obstacle for a man to marry a woman with whom he committed Zina and she became pregnant from him; to conceal her sin, and the child is his child.''
The view we adopt here in Islamweb and the most preponderant opinion in our view is the view of the majority of the scholars that the child is not traced back to the fornicator (or adulterer) at all and he does not inherit him at all; for more benefit, please refer to Fataawa 97254 and 90012.
In any case, in our view if a person considers the other opinion to be more preponderant, and he is inclined to act upon it in order to avoid a greater harm that cannot be repelled except by this, then he may act according to this opinion because prominent scholars are of this view.
As regards the issue of documents, then this depends on the difference of opinion on this issue, in cases when we stated it is permissible to trace him back to the fornicator then the matter is clear, otherwise documents should be issued for him while tracing him back to a general name, like naming him Ibn ‘Abdurrahmaan, or Ibn ‘Abdullaah, by this we mean that he is the son of one of the servants of Allaah (the Most Merciful), in which case he would inherit his mother and her family –if she is a Muslim-and the other rules of being a son would apply to him, because he is really her son.
For more benefit, please refer to Fatwa 102092.
Allaah Knows best.