Is the current relationship between the questioner and the daughter of a man with whom her mother had a past relationship, a relationship that aims for marriage, considered a "debt to be repaid" or "betrayal of her household" that necessitates accountability, and is this relationship forbidden?

The mother's past relationship does not necessitate reciprocation. Repentance is obligatory if one transgressed Allah's limits with the girl, even if it was through conversation for no need. There is no evidence for the parents' right except what Al-Ghazali and Al-Haytami mentioned regarding the establishment of a right for the female adulterer's mahrams, and that repentance depends on seeking their forgiveness if there is no fear of fitna. Otherwise, one should humble themselves before Allah, because zina brings shame upon relatives and defiles the marital bed. It is worth noting that the mother's disclosure of her past romantic relationships to her son is strange, and it is obligatory to conceal one's sins and not openly declare disobedience.