It's possible with Mastodon (import/export feature), however it's up to the service in question to support such a system. Lemmy doesn't support this yet but I believe it's a planned feature.

I self-host a Synapse (Matrix) server on a VPS and it's been working great for well over a year now. I do so via Docker and it seems very stable.

I use Caddy in production with a PHP-FPM WordPress Docker container in the same way as you describe. I have the site root mounted as a volume on to Caddy container at /var/www/html and mounted in the WordPress PHP-FPM container under a separate path. In my Caddyfile I use the root directive as you've shown but then override it under the php_fastcgi directive to match the path in the WordPress container.