I just talk how I wanna talk and ignore the haters
True, that people don't say those last three things much...so hearing them would sound a bit odd.
First off, while a puppy is a young dog, "kitty" and "bunny" are not terms for young animals, they are simply nicknames for the animals, regardless of their age. Also, the first three contain words ending in -y, whereas the others don't. And cats, dogs, and rabbits are common pets; the others are not. "kitty cat", et al. are terms of affection. We don't seem to speak affectionately of animals we don't often keep as pets.
So that may be why.