"Tubby's Drive In" was actually an A&W Root beer that was closed down. It was located (then) way out on Van Nuys Blvd. Nothing worked, no bathrooms, plumbing, grills etc. Many people sought to buy the place after the film, and convert it back into a working drive-in for a gathering place for hot rods, but it never happened.
Duke's yellow '57 Chevy was "Popular Hot Rodding" magazine's famous "Project X" test car. The transmission was damaged during filming (you can see the smoke roll over the top of the car when the engine gets hurt during the race with the Cobra).
When the filming reached into late September, it became cold enough to require the actors to keep ice in their mouths before speaking their lines to prevent frost from forming when they spoke their lines.
When the crew of Smitty's Speed Shop is gathered around unveiling the Ford, Duke (Tony Danza) asks, "Is this piss yellow?" The answer given is, "It's Tony Nancy yellow," breaking character and referring to the actor by name.