Welcome to the woes that Firaxis/2k Games bestow upon us! :(
My best suggestion is to utilize Valve's Steam client to run Civ5 and to use the online game making abilities. I cannot remember if ALL mods do not work online, but I do remember most did not. If you are running Civ5 without any expansions you are running outdated. Gods and Kings and Brave New World introduce much needed balancing and bug patches. If you have Civ5, actually enjoy the game, and would like to play with friend(s) I would STRONGLY recommend buying the expansions to get the stability. Turn off features like Unit Movements/Combat and AutoCamera moving to improve stability. Running the game in older versions of DirectX might help improve graphical speed, but network speed seemed the same.
Late game will always deteriorate sadly. SAVE OFTEN.
Patches often corrupt save files so it may require you to start a new game.
Loading maps online is possible. Make sure you are using a normal map and not one from a MOD/Steam Workshop.
Hope this helps somewhat.