Is it possible to run SQLite Manager, or other XPI, outside Firefox?
There is the XULRunner project which aims to do this, but from my experience it's a pretty big pain in the butt to set up (I tried within the last month or so) and I never got it working myself.
Yes, there appears to be an alternative version that works outside the browser. For Mac and Windows. You can find it here.
XPI files themselves are:
(From the Mozilla Docs)
This is specific to the Gecko engine (used with many Mozilla Products) and thus, can only be handled it by for installation. They CAN be unpacked up 7z or some other archiving tools, but cannot be run outside of the Gecko engine.