The hole the hamster makes in the windscreen changes position, in some shots it is against the bottom edge of the windscreen, but in others it in the center of the glass not touching any edges.
Principal Skinner decides to keep Springfield Elementary School open even though all of the other schools in the area are closed. In the actual U.S. public school system, no principal has the unchecked authority to close (or for that matter, open) their school during inclement weather. That decision is made at either a County or District level, and SES would have been shut down during the snowstorm regardless of what Skinner wanted to do. A later episode addresses by having Superintendent Chalmers announce the closing of SES due to a snowstorm, which IS a decision that someone at his level of school management can and will make in the real world.
When the kids are burning books in the library, the audio has Skinner stating "Not Huck Finn! It took hours to cross out the sass-back!" But Skinner's mouth is clearly visible at this time, and his movements don't match up to the dialogue at all. It was clearly re-dubbed at a point where they couldn't also change the animation to sync them together.
The second floor of the school isn't buried in snow. So the children would've been able to leave by going upstairs.