In the dinner scene, Homer is sitting at one end of the table, then is seen sitting at the opposite end in the next shot when he says "they do good work."
When Professor Frink demonstrates his radio-controlled toy plane, he says his son is doing a barrel roll. The maneuver shown is actually an aileron roll, a maneuver in which an aircraft does a full 360° revolution about its longitudinal axis. In a barrel roll, an airplane makes a complete rotation on both its longitudinal and lateral axes, following a path shaped roughly like a barrel.
Herb Simpson is obviously quite brilliant and successful, ergo he hasn't been affected by the Simpsons gene from Lisa the Simpson (1998)(#9.17). However, he did pick Homer to design a car without having any idea of whether he could and refused to listen to his design team when they expressed concerns over Homer's ideas, and these mistakes could possibly be an effect of the gene.
The property colours on the Monopoly board are in the wrong places.
The scene where Herb finds the newspaper with the article about Homer winning the award occurs in between the scenes with Homer winning the award, and being at Moe's afterwards. It is not possible for this article to have been printed and in the hands of a homeless man across the country in the space of a few hours.