No need to carry car papers if you travelling by air.
read following with regards to duty for reimporting car. Source
https://www.cbp.gov/trade/basic-import-export/importing-car
Re-Importing A Previously Exported Vehicle - A vehicle taken from the United States for non-commercial, private use may be returned duty free by proving to CBP that it was previously owned and registered in the United States. This proof may be a state-issued registration card for the automobile or a bill of sale for the car from a U.S. dealer. Repairs or accessories acquired abroad for your vehicle must be declared on your return and may be subject to duty.
In some countries, it will be difficult or impossible to obtain unleaded fuel for your vehicle. If the vehicle is driven using leaded gasoline, it will be necessary for you to replace the catalyst and oxygen sensor upon its return to the U.S. To avoid the expense of replacing these parts you may obtain authorization from EPA to remove the catalyst and oxygen sensor before the vehicle is shipped overseas. The EPA telephone number for these authorizations is (202) 564-2418. When the vehicle returns to the U.S., the original catalyst and oxygen sensor will need to be reinstalled. However, you may now reenter your U.S. version vehicle into the U.S. without bond, upon your assurance that you will have the reinstallation performed.