We rescue quite a few cats - from kitten to old and abandoned, the last one a very old cat that laid down on our driveway to die, completely starved. She had been declawed, meaning whoever put her out was a cruel death sentence. We rescued two abandoned kittens found in fire rubble. Another was a kitten we heard at night crying in the distance night after night, finally coxing it in. The vet estimated it was 6 weeks old and we had been hearing it for nearly 3 weeks. It's stomach so big from worms it was larger than his whole body. He survived. Looking for a missing cat at a shelter, we adopted a look-alike they had been putting off euthanizing because he was so vocal.
Dogs, though, are difference, because dogs behavior tends to be very exact to their breed and they tend to be engrained with certain behavior at a very young age. Thus, we want to know the breed of the dog and we want to raise it from a puppy for behavior reasons - this particularly so because of children. That does not necessarily mean a pure breed, because known mixed breeds can provide some fairly predictable balancing. We already have plenty of animals, plus feed hundreds of critters every day too, including a pair that are endangered.
But it is very unlikely we would adopt a dog from a shelter unless a puppy because we couldn't really know what we are getting in terms of potential problems. Because of children, one nip of a child from a dog would likely be a death sentence in effect as we wouldn't keep it after that. There was someone else's dog they left with us "for a couple weeks" and never returned. A beautiful, big dog. Very well behaved, until the day he took a very aggressive posture towards a child, basically cornering the child. We did find him a great home, but if we couldn't he was doomed and had been chained by us from that moment until we found him a home. The clock was ticking on doing so, but it worked out for him.