I don't get why cats are extinct but dogs aren't. Cats are engineered as the perfect hunters of nature while dogs are just stupid scavengers.
Actually, dogs are descended from wolves. If they get in a pack, regardless of breed, the old instincts kick in. If there are several dogs in your area that hang out and you find a that a lot of cats have been turning up dead, it was the dogs. Pack mentality. Also, dogs are actually much more intellectually capable than cats, despite that cats have that look on their face like they know whats going on. Cats would have a terrible time in a post-nuclear world, being that they only hunt small and furry mammals and the occasional reptile, the would get killed off by their own prey (mole rats and geckos). Dogs in a pack can be deadly depending on breed. Don't get me wrong, I like cats but they wouldn't make it after a nuclear holocaust.
And to the OP, horses are extinct in North America according to the wiki, and all of the snakes in the world are actually Argonians in disguise who teleported back to Nirn just in time.