First, I make reference to this topic on BGG:

https://boardgamegeek.com/thread/1165609/secrets-stone-permanent-curse

It is not clear for me if this answers are only opinions and interpretations, or are somehow offical. Over than that, I have another question. Can heroes discard by any means during their turns the Persistent Curse (by Curse Doll, or any Class skill)?

On another topic, heroes, when finding the unique search token, they carry it. If they are defeated, the token drops. Can they take it again? Quest rules say nothing about it, and general rules don't talk about it, only objective tokens which this is not. So, once they drop it (after being defeated), they cannot do anything but kill Ariad ASAP?

It's not clear if the persistent curse can be discarded- the rules seem to make it simple- if you fail the test at the start of your turn, you're cursed until the end of your turn- I don't know that there is anything that can be done about that.

You're right that the rulebook really only talks about objective tokens, not unique search tokens in regard to picking them up from the map. As written, once the search token goes down, Ariad is the only one that can pick it up.