I think most of the time sound is done before animation. In lipsyncing, generally the sound occurs 3 or 4 frames before the action.

In the a complex scene like you mentioned, you may want to import each soundtrack on by one and sync its action, "layering" actions as you add each new sound. So do character 1, layer on character 2, layer on hits and so forth. It's one of those "slower is faster" type things.