Installing updates in the background requires special permissions that require google's blessing. For now, the workaround is a Magisk/Xposed/Lsposed module to give those low level permissions to the F-Droid app.
Not anymore since Android 12. As far as I know, if the installation (or update) of app A through app B is approved once, app B is now free to update app A in the background without a prompt.
Yes, but F-Droid isn't on the play store so it wouldn't matter if they get Google's blessing to upload the app to the Play Store. They can just distribute the app with background installing capabilities from their website like they already do.