There's no reliable way to detect a proper Firefox shutdown, like the File->Exit being clicked. The only way is to monitor tab & window closing. A user can keep closing tabs until the last to shut down Firefox. In the first case, a save at shutdown would save just a blank session. Session Boss offers an alternative to serve the same purpose. It auto-saves a session whenever there's a change to a tab's URL (throttled with a 15 seconds delay), effectively handling the need of saving at shutdown. I've added some description in the "The current backup session" section in the extension info page.

For the bug, it looks like an interaction between two apps' behaviors. I'll take a look when I get a chance.

Unless there's a standard for session data exchange, it's unlikely to import from other apps' internal data files. There is a site https://www.jeffersonscher.com/res/scrounger.html to extract links from the old sessions files.