What is the origin of the word 猫 (cat)? I keep on thinking that the way it is pronounced (māo) seemed to mimic the  noises a cat makes (meow!). The character seems identical to the one for the same word in Japanese. Was this the original character that the Japanese language piggybacked off of?
The Japanese character you're referring to is 貓. Currently, in China they use 猫. Although 猫 is considered the simplified character of 貓, 猫 is already mentioned as a variant form in the 廣韻 dictionary, which is now a millennium old (although of course, 豸 is used for felines and 犭for canines...)

The question about the sound is interesting, and you could note that the sound cats make, meow, is written with the character 喵, pronounced miāo. Looking at the Kangxi dictionary, we can see that the pronunciation of 貓 also used to be some sort of "miao".

Interestingly, dictionaries don't seem to comment on the fact that they named the animal after its sound. The only explanation of the name I see is: 鼠善害苗。貓能捕鼠，故字从苗。, i.e. "Mice damage crops (lit. "苗" miao, sprouts), cats can catch them, hence the character deriving from 苗", which of course seems a bit bullshit. :)