Aphelios, Exalted has an ability that triggers when I attach an Equipment to him and tells me to “choose one that hasn’t been chosen this turn.” If I attach Svellsongur to him, does he have the second copy of his ability in time for both to trigger? And do I have to choose different options?

He does have the second copy of his ability in time, and it will trigger. The “one that hasn’t been chosen this turn” restriction applies separately to each instance of his ability. So you could choose the same option for both, but you’ll have to track which options you’ve used for each copy of his ability.