I've never turned off the camera to change lenses. Hell, never turned it off for anything (except for the tethered lock ups that occasionally happen when tethered to LR). Never had any problems with dust. And, I've changed lenses indoors, outdoors, in the rain, in the snow. I don't use caps, except on the rear element - I throw the front caps into a drawer and they're only seen again when trading lenses. I keep a Heliopan UV on the front of all lenses.
Dust on the sensor from a static charge? Never seen it, there is a shutter and mirror in the way. Of course, I wouldn't try it in a dustbowl, and I only take 3 seconds to change lenses at most. The way the contacts are laid out prevents any shorting or power application until grounding is completed anyways, so that won't hurt anything.
Now, with that said, if you have an IS lens that you're taking off, I still don't power down, but I do let the IS unit go to sleep and lock in before removing the lens. If not, you get the loose element rattle from the IS unit.