I just want to make sure that I we play this correctly. I can use my characters' abilities even when they do not participate in a challenge, can't I?

For example:

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Arya Stark

Any phase: Kneel 1 influence to choose a character with stealth or renown. Until the end of the phase, that character loses those keywords, and Arya Stark gains any keyword that was lost in this way.
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If one of my opponents initiates a challenge against another of my opponents and declares a character with Stealth as an attacker, can I use Arya's ability and pay 1 influence to make this character lose this keyword? It may save the game if the defender has only one character in play and otherwise would not be able to stop the attacker. I assume that I can use her ability, since there are no restrictions not to do it. However, I would like to have someone confirm this rule. It applies to many other cards and has a huge impact on the game.

Yes, you can always use Arya´s ability if player actions are allowed. This applies for all cards, if the card itself doesn´t bear a special requirement you can always use teh card when player actions are allowed.