While Ernest is cooking chocolate, the arm holding up Celestine is level with his head. After the cut, the arm is suddenly below his face.
When The Grey One picks up Celestine's drawing she uses her left hand to grab it at its right side. That should result in it being upside down when she looks at it, but it is upright.
When Ernest and Celestine first listen to the radio, he turns it off using the right knob. After she's left, he turns it back on and off via the left knob. Later, during the spring thunderstorm, she uses the right knob again.
When the stolen van's camouflage paint job washes off and the van rolls back into town, it leaves a trail of paint that leads the police right to Ernest and Celestine's house. The rain, however, should have washed away the trail of paint, too, and left no way to follow it back.