No they don't. Deck has some list of ordering somewhere, let me have a look around.
EDIT: Or that, deadfox
Yeah, as TheWolfe pointed out, that's in-game order, not ASB order, and when effects happen has been codified (albeit not completely) in ASB and there is no given order within categories so it is not unreasonable to believe they all happen at the same time (and can cause some interesting ties as a result).
Anyway TheWolfe, since all other forms of passive damage happen after action, I would say that Curse damage most likely also happens after action.
