Some do. Some don't. My dog and my cat cuddle up together in the doghouse. And my dog is a 4 year old Great Dane. Very big. Dogs will go after anything smaller than them with legs and that will run. Dogs love challenges and cats seem to give them.
They don't, it's more of a prey drive. Just like cats and mice don't normally get along unless they're conditioned to. A dog is pretty much alpha predator, but a cat is easily prey for dogs, foxes, and coyotes. Hence why the cat runs, and the dog chases. There's no "hate" in there at all, it's just instinct.