Kätzchen und Hündchen
So in English we can say both "kitten" and "kitty". Kitten is a pre-adult cat. Kitty is just a diminuitive implying liking, cuteness, etc. The cat may be adult, or not. (Or of course "cat" refers to any cat at all, but especially an adult cat.)
I think but am not sure, that Kätzchen may serve both functions in German. You might be referring to a kitten, or you might just be expressing liking, cuteness etc. Ditto for Hündchen.
Is this correct? Or are these words strictly for underage animals. And if so, what would a German speaker say where an English speaker might say "kitty"?
You are correct that -chen or -lein diminutives can both be used to refer to something small in size / juvenile, or to express that you find something cute / adorable. And while saying Kätzchen certainly implies that you are talking about a young cat, it could just as well be used for an adult cat, especially a small one. Cats are in a bit of an awkward position here, as there isn't a seperate word for their young, unlike most domesticated animals: Hund/Welpe, Schwein/Ferkel, Kuh/Kalb etc.