No Cats in China
One of the best sources for the history of the Chinese systems is: Walters, Derek. 2005. The Complete Guide to Chinese Astrology : The Most Comprehensive Study of the Subject Ever Published in the English Language. Watkins/Duncan Baird: London.
But let me combine my answer from his work, plus some comments by my brother, who has a couple of degrees in Chinese. As I mentioned, the 60 day cycle is quite old, dating back in certain of its aspects perhaps as far as 500 BCE. That puts it contemporaneous with the development of Babylonian astrology, and before Hellenistic had developed. The animals are technically known as the branches, a component that is combined with the stem, which is the element and Yin-Yang component. The thing is: there are Chinese characters which represent the branches that go back centuries: but nobody has the slightest clue what these characters "meant" - they don't correspond to characters for the animals that are now so familiar to us.
Walters hasn't been able to find any traces of the animal names for the branches before the 8th c. CE. And he has found no indigenous Chinese references to the use of the cat for the rabbit, although he can find scattered references in Vietnam and Tibet.
The animals do seem to have been selected because their anthropomorphic "meanings" correspond to qualities of their respective branches. Some of these work in translating to the West; some don't. Walters mentions, for example, that the Asians use the rabbit as a symbol for Spring in the same fashion as the Easter Bunny. However, the western concept of dragon is much different!
My brother mentioned that there are some issues in the translation of the actual Chinese that are going unacknowledged. For example, in Chinese, there is the same gender differentiation of words that would correspond to the English words: chicken, rooster, and hen. The word used for the branch animal is universally the one for "chicken," not "rooster," (i.e., the species, not the sex), so why does almost everybody call the branch the rooster?