Ok, so on a rare occasion I have been able to run very fast on foot, the speed of Behemoth.
The reason for this though, is due to me being mounted on Behemoth on a previous map, and upon loading into a new map, suddenly I'm on foot, but I can run just as fast as the mount moved.
Mind you most often then not, the game crashes on me afterward if I do not move to another map (by which I have returned to being mounted again).
As for attacking fast, I would not know about that. Hope my rare experience might shed some light on the topic.