Insurance is an interesting question. My guess is that many of these cars don't have any insurance at all. And they don't need it since they are not driven on the highway. For lack of a better term, I'll call these the "radical rats". Basically, they are the rusted out and pieced together cars that look like they would be undriveable. And using a quick rule of thumb: if they LOOK undrivable, they probably ARE undrivable.
On the other hand there are thousands of "nostalgia" or "traditional" rats. Or as we old geezers like to call them, hot rods. These cars are driven regularly, they are totally road worthy, and they are fully insured, licensed, and safety inspected. Yet they would fit the classification of "rat"...usually because they are unfinished or simply because the owner preferred the more spartan look of the rods we had in the 50's and 60's.
Here are a few examples of traditional hot rods that easily fall into the rat category. The first picture I took at the 2005 Hot Rod Power Tour. This guy is a "long hauler" (he drives the FULL tour in this bugger and then drives it back home) and he does it year after year. Licensed, insured and ready to roll.
This second car is actually painted, but surely belongs in the traditional category and fits the rat rod definition because it is built and maintained with "whatever is at hand". It belongs to my friend Larry down in the Milwaukee area. It's driven hundreds of miles every year in rain, sleet, snow and sunshine and is fully licensed and insured.
And finally a couple of "primer-donnas" with the old school look. The coupe is one of my favorite cars ever because, for me, it brings back memories of the "look" of some of the very first hot rods I ever saw around my little home town. But the owner, Jerry, has incorporated just enough updated equipment to make this car as safe and sane as my wife's Rav4. Again, both of these cars have been fully licensed and fully insured with no problem.
My point here is that "rats" come in a lot of different shapes, forms, colors, and sizes. Some safe, sane and easily insured to their full value. And some...well, death traps that are not insured because they have no business being on the road. So when we talk about insurance or value or drivablity, I think we need to make very clear distinctions what sort of "rats" we are talking about...the "radical rats" that are basically show cars or the "traditional rats" that are hot rods in the true sense of the word because they are built to be driven.