If you decide on puppy then the breed club is where you want to look, they often have a list of breeders and some have someone who keeps a puppy list of current puppies.
Price varies by breed, but no, paying more means not a lot, really cheap and really expensive tend to be worse breeders and good ones are somewhere between (where again varies by breed)
You want KC registration - but alone it doesn’t tell you anything, there are no good reasons not to register a litter of a breed that can be registered, so anyone who isn’t it’s for bad reasons, but it’s like a V5 for a car, it just tells you it’s definitely one of those, you’d still want to check other things before deciding to buy it.
Calm is tbh a hard one, because yes some breeds are livelier than others, but mostly it’s down to training.
It’s a cliche, but a rescue greyhound would probably suit really well, if it wasn’t for the cats, but cat friendly ones are rare, they exist, but it’s not common because they’re usually ex racers... a lurcher might give you more options?
Labs are popular because they’re very trainable, but they can be a bit nutty for the first few years.
if you’re open to rescue - breed matters less because you know you’re looking for able to live with cats, calm and biddable, so it doesn’t matter hugely what breed that comes in as it’ll ge individual dogs you’d look at.