Do Animals Know What They Look Like?

Possibly because the cat recognized the dog first. Because cats have better eyesight than dogs (especially in the dark). Dogs get even by having better noses.

This is a common and dangerous myth. Dogs regulate temperature through the act of panting. Cutting a dog’s hair in summer is a bad thing to do, and can actually cause the dog to overheat as it takes away the insulating layer of fur and exposes the dog to extremes of temperature and the skin to direct sunlight. Any vet will tell you not to do this.

I don’t see how that indicates that the dog cared what she looked like. It does not sound from your story that there was a mirror nearby for the dog to see herself in. She probably was reacting to the sensation of the haircut.