The Zotero application should be started automatically, if the Firefox connector tries to save a reference.

That's not really possible. The new Firefox extension framework is much more limited and can't launch programs on your computer.

The only way we might be able to do that would be if we installed a small service that ran in the background that the connector could communicate with and that could start the full application, but I doubt we'd do that — it'd add a huge amount of complexity, you'd then have something (albeit something lightweight) running in the background all the time, and for larger libraries Zotero startup can take a while, which would make this very awkward.

The better solutions would be 1) allowing minimizing to the system tray and 2) removing most of the functionality differences between saving to Zotero and saving to zotero.org, which would allow us to remove the "Is Zotero running?" dialog when Zotero is closed and just save to zotero.org without prompting as long as you were logged in.

The Zotero application should minimize to the system tray in Windows.

This is planned, though it might not happen for a while for technical reasons. (The equivalent exists on macOS, where you can close the main window and still have Zotero running in the background and be able to save to it.)