What is the religious ruling on a man marrying a girl whose father is unknown, but who is characterized by piety and good morals, while his family rejects this? And is a "bint al-haram" considered an outcast in society and forbidden from marrying?

There is no religious impediment to marrying the daughter of zina (unlawful sexual intercourse) if she possesses good character and religious commitment. However, it is preferable to marry a woman of noble lineage. You should try to convince your parents that she bears no sin for her parents' actions, and that she will only be judged by her own deeds, and that the noblest among people in the sight of Allah is the most righteous. If your parents insist on refusing, then obedience to them and dutifulness towards them are obligatory. The one who gives the daughter of zina in marriage is the ruler (Sultan) or his representative, because she has no wali (guardian) from her asabah (agnatic relatives).