Oh wow. I'm so late. LOL.
I actually typed that a few minutes ago, but then I got a phone call and got distracted. But yeah, it screws. It's not the same one I have, but whatever... you got it worked out. Now use the snake thingy, put a hair trap down and hopefully your problems will be resolved. Also, for good measure, it wouldn't hurt to pour some Drano down AFTER you use the snake thing, just to get rid of anything the snake doesn't get.