A true "Pony Club" shipping bandage does come down to the ground, like this: http://midcalponyclub.org/images/ima.../Bandaging.jpg
Many of us cheat and simply use a stable bandage when we trailer, which usually cups around the fetlock but does not wrap around the pastern. If you opt to do a true stable bandage, it should keep the legs clean. I would also not use polos, as they are intended to stay on for only a few hours.
As to wrapping both or just one leg, I have also heard that you need to wrap both legs, even if one is injured, to keep the support consistent. However, I've had my fair share of leg lameness and my vet has never mentioned it to me. The longest I had a horse in a wrap for was a vertical wound with proud flesh on a rear cannon. My mare was wrapped for several weeks (changed twice daily, of course) on only one leg.
If anyone has any information about wrapping only one leg being detrimental, I'd be interested to read about it.