We're renting a car to drive from NC to DLR. We've gotten to where we almost always rent a car for our trips. Yes, it's an expense, but for that expense you get an almost new car to drive. You're not putting the miles on your car (to include tire wear, oil changes and just general mileage that's not added to your vehicle). In addition, know that anything from a blown tire to other car issues is the rental's problem to take care of is real peace of mind. In our case (we have a truck too) we get some savings in gas with the rental getting better mpg than our truck. I'd assume the same would be the same for the minivan for you.
Thats what I keep telling myself too. No wear and tear on our brand new truck. No car issues to worry about. I see you've got a very long drive! You are very brave.
I think my limit is 8 hours. 
My family has always traveled by car. About the only time I fly is if we're crossing an ocean or if work is paying. We've done the trip from NC to Vegas twice now and once to Arizona (It's what we do when our kids move across the country). I also road my motorcycle from NC to SD for the Sturgis Motorcycle Rally last summer. Tacking on the drive from Vegas to Anaheim will (hopefully) be a minimal addition to this trip. 



