This is a known bug in the OS (10.8.2). It affects many applications (Spotlight, iTunes, Archive Utility, many 3rd party apps, eg BBEdit).
I haven't experienced the "doesnt quit when finished" issue, but then again I can't make it that far. As soon as you launch it, Archive Utility freezes (Application Not Responding) unti it times out and quits. But, I do know the issues are related; it's the same problem manifesting itsself different ways.
The issue is known as the "Apple Events bug". Apple is aware of it. The fix will absolutely require an OS update from Apple, so everyone just has to wait. No other choice.
I do have to say that my upgrade experience to Mountain Lion has been the buggiest and most dysfunctional experience since System 8 / 8.1 back in the mid-90's. I''m on day three since install and I can't make it two hours without having to stop and repair or investigate an issue.
Right now my list of apps I need to upgrade is approaching $1000 in upgrade license fees, and I have 7 apps that apparently will never work again as long as I live (or as long as I resist the urge to downgrade to Snow Leopard, anyway). My typical costs to upgrade are in the order of $200 when a new OS is installed.
But it's frustrating when that is accompanied by System-wide bugs like this one that affects Apple bundled software. Not good.