We tried playing a civ 5 private game, but one of our players (the one who had originally started the game) disconnected and then couldn't rejoin. Is that intended, or a bug, or what? How is civ5 multiplayer intended to work?
Your wrote a lot of side questions, but I will answer what seems to be the main one:
Can a player rejoin a Civilization 5 multiplayer game after being disconnected?
In theory, yes. I've personally done this. Try reconnecting by finding your friend on Steam, right clicking on them, and selecting the "join game" menu option, it will take you right back to the game.
However, its quite possible for whatever caused the disconnect in the first place to prevent you from reconnecting. I've personally experienced this as well, due to my internet connection simply dropping. If this is the case, that problem needs to be fixed before you can reconnect.
If you are having problems with a bad connection, there is one more thing I'd suggest you consider. As of now, in multiplayer Civilization 5, you can't just save the game. The only saves you have are the autosaves. So, I'd recommend changing your settings so that you autosave every turn. Then, at worst, you can reload the last save from one turn ago.