Well, I've spent most of this afternoon trying to unsuccessfully connect to games (and occasionally successfully) and I've ntoiced a few patterns. Firstly, if i don't restart demigod before each connection attempt, i always get stuck on the "info: System: Connecting to game host..." line in the game log.
Almost all of my successful connects have come when I'm alone in the queue trying to connect (I was connecting pretty much 100% of the time with Wagnard hosting if I was the only person trying to connect). Very occasionally I'd be the second person to connect.
When i wasn't managing to connect, most of the time there wasn't an error in the log. It just kept adding new people to connect to (who i assume must have just joined the "waiting to connect" queue, because they weren't in the actual game. Seems to suggest that you have to connect to everyone in the queue as well as everyone in the game before you can actually get into the game. If people keep joining then aborting then joining then aborting etc this could explain why it is really hard to connect when there are a lot of people also trying to connect. (I was using http://demigodthegame.com/opengames.asp + irc to find out how many were in the game and how many were connecting).
The few times i did get an error, it was this.
warning: Error running ConnectionFailed script in CScriptObject at 0326ba40: ...stardock games\demigod\dgdata.zip\lua\maui\group.lua(10): Expected a game object. (Did you call with '.' instead of ':'?)
stack traceback:
[C]: in function `InternalCreateGroup'
...stardock games\demigod\dgdata.zip\lua\maui\group.lua(10): in function `initfn'
...ardock games\demigod\dgdata.zip\lua\system\class.lua(258): in function <...ardock games\demigod\dgdata.zip\lua\system\class.lua:246>
...\stardock games\demigod\dgdata.zip\lua\ui\uiutil.lua(750): in function `QuickDialog'
...\stardock games\demigod\dgdata.zip\lua\ui\uiutil.lua(950): in function `ShowInfoDialog'
...dock games\demigod\dgdata.zip\lua\ui\lobby\lobby.lua(1704): in function <...dock games\demigod\dgdata.zip\lua\ui\lobby\lobby.lua:1701>
I also often got this, which I assumed was just telling me I'd already connected to someone and so didn't need to connect again, but it could be an error. It always happened after I'd tried to join, connected to people, then more people tried to connect so I had to connect to them. Followed a Nat connection to #### etc log.
warning: CVP: Error processing Impulse command ConnectToPeer: Attempting to redundantly add peer uid=10950
Hope this is useful.