When Farmer Hogget goes outside to inject his dog with a tranquilizer, he injects the serum under the dog's skin. In actual fact, tranquilizers are given to animals in the muscle on their backs above their tail.
In the final act, just after Farmer Hoggett latches the pen, there are three long shot views of the field, two from deep in the grandstand and one from the view of Rex and Flye. In only the latter can Farmer Hoggett and Babe be seen, there is no sight of them in the shots looking out from the grandstand.
When Farmer Hoggett is shearing sheep, it is morning and lunchtime. However when he piles the fleeces on the wagon it appears to be sundown. He then continues his shearing with the sheep, and also giving Babe a chance to drive them out of the pen.
The Boss uses the phone on the fax to call about the competition, yet the power doesn't get restored to his house until he's at the competition. - Telephones do not need the electricity to be functioning in order to work. Phone lines provide independent power to keep communication lines open, even without main power working. While the fax machine would not work without power, the telephone could still be used.