I'm just an average player, not a modder!
Oh, I assumed you were one of the developers of the Rise of Mankind mod as we get a lot of modders asking for help merging BUG into their mods.
I asked if RoM used a custom DLL because if so, they can turn on asserts themselves to make it easier to find bugs. The code that does the work in Civ is made up of several pieces: EXE (application) for the graphics engine, XML for data, Python for the screens, and a DLL (application extension) for the game rules. Civ allows you to modify everything except the EXE.
BUG doesn't have a custom DLL, but I'm pretty sure RoM given all the mods it includes, many of which alter the game rules. You can ask them to provide you a new DLL to use that would tell you more about the errors. It would be up to them, it sounds, to fix the problems, though. I'm sure they'd love users to help them find problems!
Would every one of us need to do this and compare our results? How would we do it?
A DLL with asserts turned on would give you a little more information when you got a crash. It wouldn't help with OOS (out of sync) errors. Because it's multiplayer, everyone would have to use the new DLL, but only one of you would need to write down the error messages you get.
Is there some way to determine where in all the code to look?
The RoM mod is huge and contains a lot of complex mods. Unfortunately, it's going to take a lot of play-testing to find problems like this.
Are there other places that cause OOS errors?
The main reason for OOS errors is that the state of the game differs between the different players. This happens because everyone is running their own game engine on their own copy of the game state. There is no central server running the game like in FPS games or other online games like WoW.
Could having any differences in our files on the various computers or settings cause this?
Game settings shouldn't cause the problem. If your files differ, I believe Civ4 will warn you when starting a game, or not allow you to do so at all. I'm not as familiar with multiplayer Civ as I am with other games.
Have you talked to the RoM developers about these issues?