The nylon sleeve is friction fitted into the top of the tube. If you twist and pull at it it should eventually pop out. You should have something there to center the pull rod though.
Sounds like possibly the plastic is worn away and shorter than it was originally which is letting the cymbal hop up over that. Either that or you've got too many ( or too thick ) felts under the cymbal. Originally it would have been similar to the one pictured below.
If the bushing is still the original height and the bottom cymbal is hopping straight up about an inch to jump over the tube then I'd say you're definitely pushing the envelope on sloshing the hats. Not that there's anything wrong with that but you may want to look for a more modern hi-hat stand in that case.