When Kakuzu was reanimated he assumed that since Hidan is not with him thus he is still alive. Hidan is well and truly immortal that is why Shikamaru had to bury him as Shikamaru found this to be the only way to get rid of him. Thus, this is enough to prove that Hidan is still alive and well.
Well, now coming to Kara organisation. The motives of this organisation seem to be unknown still. Also, this organisation seems to have an enormous knowledge about ninja world, Otsutsuki, etc. They knew about Momoshiki when nobody including Naruto didn’t know about him. Also, they are the ones who have information about Karma seal. So, in my opinion, they also know about Hidan and how powerful he is. I think that Kara might have rescued him and made him join their organisation.