When Homer goes to the barbershop, the barber comments that Homer doesn't have sideburns, in the next scene, Homer has thick sideburns.
Even if that stuff that Homer put on his head did work, there's no way it would grow that long overnight.
All of the power plant executives are unimpressed with Homer's big speech because he doesn't have hair, and refuse to listen to him. Yet they are quite happy to work for the bald Mr Burns. But Mr Burns is their employer and boss, Homer is neither.
When Homer is rolling around on the grass outside the house with Bart, Lisa and Snowball II after the hair formula works, the cat is bigger than the children and almost as big as Homer in the overhead shot.
There is no logical way that Karl could have known about the Simpsons' wedding anniversary, as Homer forgot all about it and this sort of information would not be in his personal record at the power plant.