As to your last question, it's actually pretty much always running, listening or not. At least in my installation, clicking the X to close the window actually (by default) just minimizes it; to really close the application, you have to use a context menu or something. That alone is enough of a hurdle that I rarely bother to close it...on the assumption that it's not hogging too many resources. I'm wondering how safe that assumption really was now.
I do not believe it does quit it properly on OS X. Even after a fresh reboot, Spotify appeared to be using about 40k/sec outbound bandwith. After I removed it, idle bandwidth dropped to zero.
I also noticed something strange when removing it with CleanMyMac (similar to AppZapper). There's a separate dock icon file for Spotify, and removing it restarted Dock.app. I think that might be to cover up Spotify running in the background somehow.
Spotify does quit when you cmd-Q. You can verify this by opening Activity Monitor and looking at the process list.
If you are running MacOS X Lion, then the OS might keep the underlying OS process around in case you start it again -- this is by design. I have not been able to observe this personally, however.
