The sign on the schoolhouse reads "NORTHFORK SCHOOL." The name of the town throughout the series has always been two words: "North Fork."
Billy claims he won't be wearing shoes until first frost but that's not acceptable to Mr. Griswold and he will talk to his mother. Very next day when he's playing hooky with Mark, Billy is wearing laced shoes and it wouldn't happen that quickly.
Lucas has never met the new schoolmaster until that first day of school. Later Lucas mentions to Mark how Griswold was approved by the school board and will remain in the job. Lucas is the chief member of the school board however Lucas never met him.
The wire holding up the dislodged beam is clearly visible during shots showing the precarious nature of the situation in the mine shaft.