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.
SQLiteManager is a powerful database manager for sqlite databases, it combines an incredible easy to use interface with blazing speed and advanced features
XPI files themselves are:
Cross-Platform Installer Module (XPI) is a ZIP file used to install packages, utilizing the XPInstall technology. XPI modules (so called "Bundles") are employed to install a wide variety of software, including Plugins, Extensions, Themes, and Thunderbird dictionaries
(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.