No, Costume chamber is for S1. All other costumes will be in the portal for that hero.
What portal ?
Whatever portal that hero is normally in.
No; as @Shunt said the non-S1 costumes will only appear in their normal portals… So for old HotM that will be the Tavern of Legends, Seasonal in Seasonal portals (i.e francine in Halloween, Lepus in Spring etc…), S2 costumes in the Atlantis portal etc…