Last time I checked, Synapse was the only usable homeserver implementation. Has this changed yet? It's not that big of an issue, but it doesn't seem much of an open protocol if everyone is just running the same implementation.
Yes, depending on your needs Dendrite (Go impl, written by Element team (mostly Neil?)) works well (I've heard) and some people are even running Conduit (Rust impl, independent) as their main.
I run Conduit as my main. It’s still behind on features but is active and very usable. No complaints.
The real standards are IRC and XMPP. Matrix is more a product / custom protocol.