Why would cats go extinct? Cats are pretty much everywhere on the American continent and in great numbers, I would expect house cats to exist, in some form, before I would animals with a smaller range and smaller population size. Honestly, larger creatures would be way more adversely impacted -- large animals need more food and resources, which would be in small supply; and, for herbivores, that means lots of potentially contaminated plants, for predators, contaminants move up the food chain and accumulate (and in the case of cats, mice and rats are also small and wide spread, so their food source should be rather abundant).
Assuming that 10,000 bombs were dropped, unless they were dropped using an extremely extremely sophisticated model of fallout maximization (over all area, not over strategic area -- in other words, if they were dropped in a way contrary to destroying important targets), then it is almost impossible that there weren't decent pockets of uncontaminated rural land (especially if we are considering all of North America). Again, cats and mice are everywhere, so they would have been in these areas, thus, surviving.
As for House, how would he know? It's not like he can really travel about and check, or phone others all around the nation and find out -- and, really, how much of his time do you think he truly devoted to this (in other words, even if some cats were around in places he could know about, would he waste time to make sure).
Finally, Even if they survived, nothing entails that they would be wide spread; it isn't likely that cat packs would form and spread out, they would most likely cluster around nearby human habitats, spreading with them (humans civilization attracts mice, which, in turn, attracts kitties).
*Ultimately, of course, this is up to the creator, but cats are, probably, quite far down on the list of things that would go extinct in a nuclear war -- then again, Fallout nukes seem to be a little bit magical (actually, a lot), so maybe their radiation just really hates cats:p