After the musical number "Romeo and Juliet," when the Lecturer is holding the Bible and moves it out of the shot, you can hear the sound of the book landing on the grass, like it was tossed down. In the next shot, the Lecturer is again holding the Bible under his arm.
During the song "The Stuff" Ana Gasteyer looks out the window and sing the line "Rapes Me". Before she looks out the window, her cigarette is in her left hand. It then switches to the right as shes singing out the window. When we see her again back inside the window, the cigarette is back in her left hand.
While it is indeed true that FDR was wheelchair-bound by polio, he had perfected his ability to walk with the support of someone else's arm and could stand upright for a long period of time. He never appeared in public in a wheelchair, and in fact it was not even known outside his circle of associates that he was wheelchair-bound until after his death.