Unfortunately, there's a body warm-up, too. Your spine is particularly vulnerable to injury right after you get out of bed because fluid accumulates in the discs overnight. As far as I know, running is fine for people with no existing back injuries, but anything that loads or rotates your spine is iffy. You're 90% back to normal an hour after getting up, though, so if you can find something to do in the meantime without falling asleep again, you're golden.
Yep on the spine thing. A morning workout is a good way to get metabolism up after resting, but anybody training (except for marathons, which largely start in the morning) should work out at night, when the body is more efficient.
A workout in the morning now and then helps, as the body isn't accustomed to working out in that time period. It's similar to the way a lifter changes workout structure to prevent the muscle from getting accustomed to a particular lift.
