Finally got a friend playing Civ III and we've been doing MP through Stream as well. There are definitely some kinks in the process but not being able to reopen a game has not been one of them.
The most reliable process to Host MP games is as follows:
1. In the start window, click Multiplayer
2. Select Internet game and click OK [O]
3. In the Multiplayer window, click Host (opens Staging Window) to host a game
4. In the Staging Window, wait for the other players to Join your hosted game under Player Setup. Once everyone clicks the Ready box (shows a checkmark), then click Launch.
5. In the Multiplayer Setup window, under Game Mode, choose Load Game and select the game to restart.
6. Once everyone has clicked Ready, click Launch.
The saved game loads, is copied to each of the players and then starts.
Hope this helps.