apparently it has navidrome support :) navidrome enables you to have your own music library on a device from that you then stream it on your other devices

The other way around, and yes.

https://kawaiidango.github.io/pano-scrobbler/ is one example, but most decent Desktop/Mobile applications have some sort of plugin.

Depends on the app.
If it caches the plays and syncs it back to the scrobble service?
Probably possible.
Afaik it's not possible for Jellyfin.

Depends. LB supports traditional Lastfm scrobling, so you can use players that scrobbles to Lastfm if you can change the URL

The in-app player is more limited, but a good old Navidrome instance is plenty enough!