You can travel with many passports if they are valid and have at least 6 months on them. America does not recognize dual citizenship so always best to show only your American passport to them. You would require a visa for the Thai passport if you used it. So out with the Thai passport and into USA with the American. Simple and reverse coming back. The Thai passport is good in Asean countries because you don't have to pay for a visa to enter them. Hope that helps.
It's a small purple book you apply for at the Land Transport Office. Can't go outside Thailand without it. Takes a few days to a week. You renew it every year. It's cheap to get. You're actually exporting your vehicle out of Thailand then importing it back. As strange as that sounds.
If you have the passport book for the car and it is all in your name including the blue book, it's fairly easy but lots of paperwork. Insurance is a must have on the Laos side easy to buy and don't forget to tell them you will be exiting at a different border if that is the case. Drive on the other side and enjoy the people.