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. :)
you're correct on both of your assumptions; but keep in mind, simplified Chinese comes from traditional Chinese which in turn has origins. And in this case, 猫 came from 貓; the left part of the word, 豸 indicates its Cat type animal and the right side, 苗 gives it how it sounds when it makes sound. and together it described what kind of beast it is and what sound it makes.
and if you see kanji in Japanese you can almost always assume its coming from Chinese
This is a pictophonetic word. It is originally from the Traditional version of from the word: 貓 豸 suggests that it is some sort of animal. It actually means legless insects. That's probably because of the way cats move (slyly, and smooth)
苗 provides the sound of the word, which is "Miao". It plays no part in the actual definition of the word because this part of the world means seedling (with regards to Grass - 田). You'll find that many Chinese words have some sort of agricultural root in them.
So yes, to your point about the sound that the cat makes, the originator of the word found a particle that match the sound, and then appended a particle that signifies an animal or creature.
The Mandarin word māo may indeed be onomatopœtic — many dialect forms have tones and vowels that are inconsistent, showing that as a group they represent onomatopœia.
But the character (貓/猫) has pretty clearly represented a word for something feline for a very long time. The Shījīng's "Dàyǎ" has a line "有貓有虎 [there are cats and tigers]"; the Lǐ jì has a line "迎貓為其食田鼠也 [the reason they welcome cats is that (cats) eat fieldmice]."