When a character moves on to the path of Herman when he has Hermetica Occulta equipped, who is initiating the duel? This caused a disagreement between me and a fellow player. I said Herman since it is his equipment, they said the other character since they moved on to the path.

Since the duel happens during the opponents activation, I'd say the opponent initiates the duel. Furthermore, the rules text of the Hermentica states that the opponent has to win the duel and that each opponent may only initiate this duel once per activation. So it's quite clear that the opponent initiates the duel.

Yes, I agree with ??!. The rules are quite clear on this point.