I've made a number of 1400 mile trips with my cats. Neither are alive now, and I've decided not to get another cat until I stop making these drives next year, but I generally didn't have any problems.
I used the plastic airline carrier cage for each, but I agree that a more open cage would've been better. I put them in the middle of the back seat, wedged or somehow buckled in for safety, and so that they could see me, and made sure the sun wasn't beating down on them. Absolutely, positively, would not drive with a cat on the loose in the car, for the already given reasons of it not getting underfoot, and the chance of it escaping when I got out. I put a towel in there for comfort, and to soak up any "accidents". There was one time one cat pooped as we were going through Beaumont, TX. It only changed the smell, didn't really make it any worse!
At rest stops I'd let them out inside the car, with doors shut, to see if they'd use a litter box I'd bring along. They rarely did. I might've offered them a bit of water. I always broke the drive into two days so they wouldn't stay cooped up too long, and researched motels that allowed pets. Always parked in the shade for any stops. If I couldn't for gas, I'd move it as soon as I was done. One cat would howl for about the first hour, and that was it. If it had been non-stop, I'd have considered tranquilizers for the next time.
At a new home, my only concern was that they be able to find the litter box. A few times the first day or two I'd "fly" them just above ground level to where the box was, trying to give them the same view and scents to find the box again. Never a problem.
Butterred cat paws? There's a new one! Sounds so crazy it just might work!