Facet Joint "clicking"
Complex stuff there.
I tend to ignore clicky joints, some people just have them, and they can develop at any time. If it increased after manipulation, then I still wouldn't be concerned; if it stayed increased after a course of treatment and resolution (including rehab) of the original problem, then I'd start taking notice. I don't really see how it would be related to hypermobility though (but haven't looked into it at all either).
Hypermobility can indeed be misdtaken for hypermobility; because even a hypermobile joint can become transiently hypomobile; in which case it needs movement added, but you want to be careful of how you do this, I generally favour much more gentle procedures (mobilisation) when I suspect it.
Hypermobility is extremely difficult to find though, it reacts normally to challenge after all - there are tests for global hypermobility, but they're pretty poor, and not terribly relevant.
Generally speaking, I treat degenerative joints and pregnant ladies as potentially hypermobile; beyond that, it's an impression, a suspicion only; certainly not something you can really quantify.