No. There is a cartoon called "Cat-Dog". But, there is no cat-cat or cat-dog in reality.
Well, my cat has been chased in by our other cat. so it is possible for that to happen to yours.
racoon
Some well-known diseases in cats are Feline enteritis (Panleucopaenia), Cat flu, Feline leukemia virus and Feline immunodeficiency virus (FIV). Domestic cats should be vaccinated against the first...
If you mean Cat in the Hat, the author is Dr. Seuss.