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"home server" is the term for the server your account lives on in Matrix, not for "server you run at home", although you can of course do that.

The federation requires a publicly accessible server to work. I got a cheap domain name, pointed it to my public IP (with DDNS), and port-forwarded to my server. Works fine for me.

DNS is required for wildcard domains.

To issue a certificate for a specific subdomain, you just need to have an http server and serve the secret under the .well-known directory.