A. Yes, you do have to pay income tax on the fair market value of the car.
The charity will issue you Form W-2G with the value of the car, and you will report it as "Other Income" on line 21 of your Form 1040.
Even you sell the car at below the reported value, you still have to pay the reported taxable income tax. You can check websites such as KBB.com and Edmunds.com to find out the true value of the car and share that information with the reporting institute to influence its decision on the value.