I'm no doctor, but I'd say any exercise is better than none.
Not atall, alot of bouncers work out at the gym near my bit late at night, you could be more tired but because of the long day, as long as you eat something before (one hour before it) and after the workout you should be fine with energy
it will or can keep you awake,try it and see.
It is bad, if You wish to sleep, for exercise wires you up.
Yes. It makes sleep difficult because your adrenalin's still pumping.
They say you should have a few hours in between bedtime and exercise.. I'd say what two hours? just a guess