How do I do a network game? This would be with someone in Australia (I am in Ukraine). We have tried each both as host and client to start a server through GOG and DosBox, but the tunnelling always fails on someone's end. How do I fix this?

I am also trying to find an answer to this. I have tried to join a friend's hosted game & it will ask to provide a host IP address... I can type in my friend's IP address but the connection fails despite using the same node.

Are we required to have a VPN? Or is it a basic 127.0.0.1 server IP address that we are magically supposed to know?

There is instructions on GOG site (article for game and general articles for DosBox) and on DosBox wiki, or PCGW Wiki (they all have articles dedicated to DosBox Network play).
Client should put host external(*) IP address.

* You can use GameRanger or Radmin VPN. It be differ IP, provided by client. As for DosBox... it is also IP provided by DosBox, right? From it virtual IPX-IP service... Must recheck wiki'es myself.