When in order to activate an Action, you have to pay a cost (Pay 1 to...), are you draining the resources following the same rules as if you were activating a card (i.e. you need at least one resource matching the faction of the card you want to play).

As an example, if can I activate a Hastur character's ability draining a domain without Hastur resources?

If you refer to the "Hastur King in Yellow " ability, it only specify that you need to drain a domain with at least 3 ressources on it, no matter what faction's ressources you've put under the domains.

If a text box don't say that you need to have specific ressources on a domain to play a card (look at the FAQ, section steadfast).

Sorry I did not explain myself well.

Let's have an example: Clover Club Torch Singer (F66) says: Action: Pay 1 to choose a character. Until the end of the phase, that character gets -2 skill.

My question is: In order to activate the card's action, do I need to drain a domain with at least a Syndicate resource as if I were put into play the card or can I drain a domain with no matter what its resources, as long as I drain one resource.

No, there is no need to make a resource match when paying the cost of a triggered effect (Action, Disrupt, Forced Response or Response) any resource will do. So in your example, you could drain any domain you control (with at least one resource of any type on it) in order to trigger the Clover Club Torch Singer's effect.

Thanks for the reply. It wasn't obvious reading the rules (unless I missed that part).

Another question: Can you trigger the effect of a character who is exhausted or insane?

Regarding triggering effects on characters-

Exhausted: Yes. As long as exhausting isn't part of the cost, i.e."Action: Pay one and exhaust ...."

Insane: No. Characters who are insane have their text boxes blank, along with their name, cost, type, skill, icons, etc. As that is the case, they have no longer have text telling you to activate anything. The only thing that would potentially be activiated is something along the lines of "Response of Forced Response: When NAME goes insane.....". That would still trigger upon them going insane.

Thanks for the clarification.