When the cyclists are racing at "Monumental National Park" (actually Colorado National Monument, west of Grand Junction, Colorado) one of the cyclists (an extra) falls off and goes over the edge. He actually went over a drop off that was about 1,000 feet, and was only saved from major injuries when a small tree snagged his bike.
Muzzin's nickname is "The Cannibal". "The Cannibal" is also the nickname of the real life five-time Tour De France champion Eddy Merckx who also appears in the film.
Naming the dog "Eddie" (in the 'training with Eddie' sequence) is possibly an inside reference to Eddie Borysewicz, who became the US national cycling federation's first full-time coach in 1978. Known as "Eddie B" to his riders, he was responsible for America's rise to cycling prominence, starting with the 1984 LA Olympics, in the 1980s.
In the "shake and brake" "wheelsucker" horse race scene, the cowboys were told to hold the horses back, as the director wanted the cyclists to beat them.
When athletic supervisor Dr. Conrad and his lethargic, overweight son are discussing the kid's intention to become a bowler, the boy, Randolph, mentions the lack of black athletics in particular sports, like hockey, and that there's only one black golfer. He was referring to Calvin Peete, who was the only African American professional golfer for years.