Other mistake: During the scene where the teens are performing that "kicking" routine in the gymnasium as led by Tommy Djilas, you can see Harold Hill in the back left of the screen, watching. Right before the number ends, a costume prop can be seen coming from the back middle of the dancers and hitting Robert Preston in the face.
Other mistake: Immediately after the unexpected breakup of the 4th of July festivities at the High School, Professor Hill intercepts Zaneeta on her way to the "library." 4th of July is a national holiday and it is unlikely the library would be open since it is a public facility ("he left the city the library building, but he left all the books to her"). Second, even if the library was open, this scene is less than five minutes after the festivities were unexpectedly broken up, so Marian (the librarian) would not have scheduled an opening time until later, after the festivities were expected to end.