The Apollo 10 astronauts were tempted to land on the Moon. In the real Apollo 10 mission, the Landing Module was given only a partial load of fuel. If the astronauts had landed on the Moon, then they wouldn't have enough fuel to lift off from the Moon and dock with the Command Module. This was done by NASA to remove any temptation the astronauts might have to land after getting within 10 miles of the Moon's surface.
During the Congress hearing, Von Braun's pictures with Hitler and other high ranking Nazis are real, as well as all the accusations by Congressman Sandman. In real life, the amount of his actual involvement as a war criminal, as well as his membership in the SS, were kept secret in order to maintain public opinion on his side. The song is real too: written by Tom Lehrer, it was falsely rumored that Von Braun sued him over it.
The cosmonaut lifting the gold visor while on the moon is based on what astronaut Harrison Schmitt did on Apollo 17.
The archival footage of the real Werner von Braun from a 1955 Disney TV series shows him standing beside a model of a proposed manned spaceship. The rocket shown in the series was offered as a 1/480 scale model kit by Strombecker in 1956.
Michael Collins is seen with a mustache during the picnic party. Collins grew a mustache during the real-life Apollo 11 mission and kept for some time after his return to Earth.