Is what occurred in the aforementioned two cases—the first concerning the father's relationship with a woman and the second about stories from his youth—considered a slander (qadhf) necessitating the prescribed punishment (hadd), or is seeking forgiveness (istighfar) and repentance (tawbah) sufficient?

What transpired from you and your mother is not considered a legal accusation (qadhf); rather, it is merely apprehension and doubt. Doubt regarding the form of the utterance issued by both of you nullifies the ruling of qadhf, because the default is innocence, and the default for an incidental act is non-existence. You both must repent to Allah for delving into this matter and repeatedly discussing it, and seek forgiveness for those whom you backbit.