As Mr Bloomgarden and Beaver talk after the fight, the lining of Beaver's hat is showing somewhat, and it's on his left. The very next shot has the lining sticking out much more, and it's on Beaver's right side.
Two different times Ward opens the door to exit the kitchen and there is a blank wall to his right instead of what should be the stairway balustrade.
It's never explained why Aunt Martha doesn't attend to June's sister.
Before Beaver walks into the classroom, he says, "Hi, Jane" to Judy Hensler. (at around 12 minutes)