Dylan pulls into the school and as he parks he uses the shifter on the steering wheel column to park, but when he races you see him shift gears from a floor board shifter.
In the scene immediately after the final race, Dylan walks around the corner and runs into Mr. Johnson. As he's turning the corner, there's a snow flurry falling, but it's gone during their conversation and in the following scenes.
In the movie it's stated that Fast Freddy successfully completed the "Twin City Run" thirty years ago. As the movie presumably takes place in 1989, that would mean that the race took place in 1959. When Monkey explains to Dylan everything he did to tune up Fast Freddy's 1957 Chevrolet Bel Air, he says that the car has a 427 big-block engine. The 1957 Bel Air was not factory available with a 427 ci engine (or any big-block engine for that matter) and even if the motor was retrofitted into the car, the Chevy 427 did not exist in 1959, as it was not introduced until 1963 (as a special option) and not mass produced until 1966.
In the opening race scene, Malone is seen shifting gears of a floor-mounted manual shifter in his 1968 Chevy Chevelle, but the close-ups of his feet clearly show the wider brake pedal of a car with an automatic transmission. Also, the instrument cluster of the Chevelle depicted is not from a 1968 model (its a shot of a 1970 model with the rally gauge package).
When pulling the '57 Chevy out of the trailer it is red with the insert on the rear painted white. In the car wash when the guy is using a rag on the side its easy to see that the insert is now the aluminum used on the Bel-air.