I can confidently say that no it will not “intelligently use the LAN” when you’re on the same network - I don’t know of any service that will… unless Plex/Plexamp somehow does this?
The solution is as someone else said - use a DNS Server to forward it in your LAN to the internal IP. If you’re unsure how to do this, just search how to setup a Hairpin NAT for the router you own. I can confirm that once you set this up, it will work seamlessly with both Finamp and Jellyfin.
Gapless? Do you mean downloading media for offline playback? Yes:
Just be prepared for the space requirements of your media library as you may find your phone quickly running out of storage if you have a lot of high res audio: