Re: Claws of Kokytos, how to defeat it.
 

I think it goes away next time it decides to attack someone, but I'm not sure of that.

Re: Claws of Kokytos, how to defeat it.
 

I suspect this is not true, but I'm not sure. I do know that I've found one sitting in a province after banishing it (was likely to actually have been fire version of the spell, but don't see that that makes a difference), which under your hypothesis would only have happened if he appeared in a totally empty province.

I think I successfully tried to Hellbind Heart it and it did disappear after a short time, but that may simply be a property of the creature... IIRC it's even in the text description of it, but I don't recall, it was a year+ ago. Will see if I can repeat it in my current Mictlan game.

Re: Claws of Kokytos, how to defeat it.
 

I meant the next time it appeared to assassinate a heavily Horror Marked target.
I would expect it to stay in the province and fight all comers until then. Though, as you say, there may also be a random disappearance property.