matrix-org/synapse ? reverse-engineered prompt

Build me a self hosted Matrix server like Synapse that I can run for my own chat service. I want people to be able to connect with Element or another Matrix client, log in, create rooms, and send messages, with federation working so it can talk to other Matrix servers too.

Please make it feel production ready, with a simple Docker based setup, sensible default configuration, PostgreSQL support, TLS, and a setup that works cleanly behind a reverse proxy like nginx or Caddy on port 443. Keep registration off by default, but make it easy to turn on later with some kind of protection for public signups. I also want clear install and upgrade instructions, a sample config I can actually edit, and a few troubleshooting notes for common setup problems.

Please use safe defaults around domain separation and media hosting since this is a public facing server. If anything here looks old, check the current Synapse docs online and follow the modern approach.