I wonder what difficulty settings people play on, as various accounts of two-bridge battles rarely match with my experience. I always play on hard or expert, which I know gives the AI a bit more tactical nous on the battlefield, and I generally find that if the AI is defending a two-bridge map, then both bridges are covered by its forces. Admittedly on one (usually the remote one from the starting position) the force is small, maybe a single spear unit plus a single missile unit, but it is very rarely, if ever, left undefended, and the main army will move across if it looks like I might be trying to take that bridge (hence the advice above about running fast cav between the two bridges )

In attacking though the AI doesn't seem capable of using the second bridge to try a flanker, even on expert setting. Even so I do still tend to keep it covered, for that one day when I rely on it...

As for organ guns, there was a thread dedicated to their uses or otherwise. I generally find though that in bridge battles, even when I've had them available, I've tended not to deploy them. The main reason being that there were enemy artillery pieces on the opposite bank. A catapult out-ranges an organ gun and can splat it easily, and a ballista could pick off its crew safely. Even arbalests can reach an organ gun unless it is one of those long bridges which are so rare. Now, if I was defending against an army composed of Swiss armoured pikemen then I would deploy the OG, but with any sort of serious (or even half-serious) missile component against it I would probably save it for another day.

As for the other types for use against men, I think the serpentine generally wins on accuracy, certainly once it has a bit of valour, it will usually hit the unit you aim at, whereas the culverins etc will occasionally land a shot somewhere close enough for the target unit to hear the bang, but generally they needn't fear for their lives....