You can also try one of those cheap plastic snakes if chemicals didn't do it.

Yeah that coupling ring is threaded and will unscrew. Make sure you are trying to turn it the right way - the ring in the 2nd photo will unscrew downwards and stay on the elbow.

Don't worry too terribly much about breaking something - you've got a couple more fittings before you lose the pipe to the wall. Should be easy replacements if something does break.