YES there is noticable input lag
NO it's super fast compared to other games
I can't tell
I notice it on Xbox One but not on Xbox 360. I think it has to do with the dead zone on the Xbone controllers as others are saying. I really hope Bungie provides a way get the same experience on both consoles. For now I'm sticking with X360 because even though the gfx aren't as good, the lag is too much to handle in MP.