I play MP only, however I play with friends and we don't play it as it is designed. While we are playing MP we all are online using voice communications on a ventrillo server. When war is declared the players who are at war move their military units as if it were in turn-based mode, with the player who declared war moving all their units first, then the other player.
This doesn't take all of the simultanous turns out of the equation. Earlier numbered players in the game still have an advantage in terms of wonder building. Camps/huts can sometimes go to whoever clicks first. We also play with AIs, so sometimes things can get tricky if two players are hitting the AI and wanting to take the same city, but this is rare.
With regards to playing with the AI - I am confused that people complain about how it handles its diplomacy. The only difference between MP and SP with regards to AI diplomacy is that the AI won't contact you for deals. It'll still declare war on players, on other AIs, and still have the full diplomacy modifiers. Now of course this means you don't get contacted to declare friendship/denounce/any of that other stuff, so it's a little different, but I don't think it's really significant.
Connection problems are rarely an issue for us. We often get large lags at the start of turns on big games near the end, but this seems to be due to one person's computer not really being up to handling things as quickly as the others. We will occasionally get desyncs, which does seem to give the human an advantage over the AI as it seems to skip the AI's turn, but these are fairly rare, especially in the early game.