If the artificial intelligence from emulation is accomplished through tweaking an emulation and/or piling on computational resources, why couldn't it be accomplished before we start emulating humans?

Other primates, for example. Particularly in the case of the destructive-read and ethics-of-algorithmic-tweaks, animal testing will surely precede human testing. To the extent a human brain is just a primate brain with more computing power, another primate with better memory and clock speed should serve almost as effectively.

What about other mammals with culture and communication, like a whales or dolphins?

Something not a mammal at all, like Great Tits?