Does what the husband said to his wife: "Were you going to college to study, or were you doing such-and-such with the young men at college, and your former fiancé, you'll find he used to kiss you and hug you and do everything with you," constitute an accusation of illicit sexual relations warranting the prescribed punishment for such an accusation (hadd al-qadhf)? And what must he do if that is the case?

Explicitly accusing someone of adultery necessitates the prescribed punishment (حد القذف) by scholarly consensus. The words you mentioned to your wife are not explicit accusations of adultery, so they do not necessitate the prescribed punishment unless you intended to accuse her of adultery. The prescribed punishment for slander (حد القذف) is a right of the slandered person and is waived if they forgive. It is not carried out except upon their request. You must repent to Allah and seek forgiveness from your wife.