[…], from that point the app will be built by f-droid with their own digital signature.

This part of your comment is not quite true. One of the advantages of reproducible builds is that the app can be signed by the developer but fdroid can still verify that it has been built from the correct source code. You can check out the documentation here: https://f-droid.org/docs/Reproducible_Builds/

Rainbowdragoneyes makes good chiptune music (no fusion). Very underappreciated, it feels like.

Edit: added link

Unless the app allows you to export/import your settings, unfortunately the answer is no.

This is because F-Droid builds its own installers and signs them with their own key. Android (sensibly) does not allow upgrading a package if the signing key is not the same.

Mullvad DNS works well for this too: https://mullvad.net/en/help/dns-over-https-and-dns-over-tls/

Using DoH (DNS over HTTPS) means it should work on networks that try to block usage of private DNS servers.