1. Yes. Carpet backing is very abrasive. When you carry a roll of this stuff around, wear gloves and long sleeves. I am wondering why you are asking about car paint. Don't carry this stuff around on the top of your car.
2. When I pull carpet, using a FRESH utility knife blade, I cut strips about 5 feet wide and 10-15 feet long. I roll it up with the backing on the outside, then use duct tape in a single wrap all the way around the roll, keeping the roll as tight as possible. As far as dust and dirt, forget it. This is a messy job. Period.
3. I don't install carpet, so I don't know about ease of installation, but I do know that to cut out and lay out rooms, an installer usually rolls out carpet on the driveway or garage floor, so it works better to keep the carpet clean by having the backing side down. So, having the roll with the backing side out is probably helpful from a clean product point of view.
Beware of carpet strips. They are very very sharp. Use a prybar or a flat shovel to pull them out, and watch your fingers!