Yes, this is what I do. You just set a DNS record on a subdomain, e.g. matrix.mydomain.com, and use that as your home server. Your username can still be @me:mydomain.com with this setup.
This is a good question, as historically different protocols used different ports to communicate, meaning you could have many services on the same domain/IP, but recently many new protocols run over HTTP port 443... And I think Matrix is one of them, so the answer is no, you cannot host both a website and a matrix server on the same domain name. Happy to be corrected if I'm wrong.
Actually, it uses port 8448 for federation [0], which is unlikely to conflict with other services. But, even if it does, there are ways to specify a different port or subdomain [1].
No, you can have both. All you need to do is proxy a few paths to the running instance of Matrix. Those paths are matrix-specific and won't interfere with anything else. I've been running on this configuration for some time now.
Either add an DNS SRV record to example.com pointing matrix to matrix.example.com, or server a single JSON under .well-known pointing matrix to matrix.example.com.
In my instance, my root domain is served from CloudFront, so in this instance I could add an A record to a homeserver VPS and use a SRV/.well-known to point to it :)
