We use a photo booth at this store and our Hyundai store. Theirs is awesome, but we've evolved ours into something good (and certainly won't get our work confused for any stock photos!).
Space: We have two converted service bays to ourselves (well, normally). This gives us a 25x25-ish space to work with. This is great for anything up to about an Acadia or Enclave, tight for Yukons and Sierras, and suck-in-your-waist territory for Sierra 3500 trucks. Flaps in the background material provide access to neighboring bays, the bay doors to get the cars in and out, and the all-important bathroom. One corner holds a desk for a computer and printer (we do stickers and entering features at the same time). Through well-tested camera angles, we avoid showing much of anything.
Lighting: We have a mixture of light sources--I can hear my photography professor from school cringing, but it works well enough without resorting to paying the [strike]extortionate[/strike] prices of professional lights. For lack of mouse drawing skills, I freehanded a rough diagram for you:

I'm still not 100% satisfied with it, but it's where I can make up for it in post (now my professor's probably having a heart attack). Since most of the light's pointed away from the backdrop, the nearly-white nature of it doesn't really pose a problem.

On the interiors, we shoot exclusively using the on-camera flash. Even for the odometer shots. How we get those to look right is a bit of a secret, though not one that's hard to figure out if you've got a general grasp on using flashes. (Since you're bringing up monolights and softboxes, I imagine you're already dissecting this photo.)

Process: Back the car into the bay, shoot the side and front, then go and do the interiors. Pull the car around, shoot the back. (If I had more room, this would probably be superfluous.) Punch features into eBiz, print and apply stickers (for used cars), get the car out, roll on to the next one.
You're dead on about the necessity of having the right hands handling the process, especially running the camera. It takes an hour to learn, and a lifetime to master. (I've been the only shooter here for 18 months, and I'm still tweaking the process. If I looked at some of my early output, I'd probably smack myself over the head. Yes, even before my photography professor did.)
Best of luck!