Welcome to the RTA Forum!

Honestly, I think you are seriously overly worried about your budget. You've got a little over $100 a day, and that's after you've accounted for fuel costs. While you can't go wild with that amount, that's not a rice and beans budget either. If you stayed at Budget hotels every night (roughly $50 a night), you'd still have more than $50 each day to eat and cover all of your extras.

As far as where I'd suggest you sleep in your car - I'd suggest you not even try. Proper rest is essential to having a good roadtrip, and that's just not possible with two people inside a car. If you insist, don't even consider rest areas - an extremely dangerous spot to spend the night. Truck stops are safe, but well lit and often loud - which leaves campgrounds as your best choice. However, investing a relatively small amount of money in camping gear would be much wiser than trying to sleep inside your car.

As far as places you "need" to get a motel, that's completely up to you, but if you're looking to explore the nightlife in any city, that's a very good place to start.

There's almost too many National Parks to list along your route, and a quick look at a map will point them out to you. A very quick list - Sequoia, Yosemite, Redwoods, Lassen Volcanic, Crater Lake, Mt. Rainier, Glacier, Yellowstone, Bryce Canyon, and Zion. Really though, browsing some maps is going to give you a much better picture too work with.