Apologies for the late response

I can access every node by IP (IPv6 to be precise).
Discovery within a local network happens through regular broadcasts. For connecting different networks, you need to set peering addresses that are reachable and configure the other side to listen.
You only need one node per network though, the others will automatically discover the path and connect on the best route to their target. If your node in the middle falls over, any other node that's reachable can be used instead. The Yggdrasil Blog posts have some explainations of the algorithms used.

There's no explicit gateway, but you can use standard routing and firewall tools to do whatever you want. I only use it for accessing internal stuff, not as a full VPN for my client devices, but you could probably make that work by setting one node as router and configure its Yggdrasil ip as you gateway (excluding the traffic you need to connect to the VPN).

One downside is that everything's still in progress and most versions change significant parts of the routing scheme, meaning it doesn't work with the previous version. It is primarily a research tool for internet scale mesh networks, but releases are also infrequent enough where you shouldn't worry too much.