I do treat animals with different levels of sympathy, although I'd like to think that I do it based more on material differences between the animals than on how I personally feel about them.
I save snails and worms from the sidewalk after it rains. I don't always save slugs, though. Part of that is because I can't grab a slug as easily as I can grab a snail or even a worm. Probably another part, though, is that I don't find slugs as aesthetically appealing.
Last night I found dozens of baby spiders in my bathroom. I let them be, but this morning when I took a shower I had to let one of them go down the drain. He was on my shampoo bottle, and I gave up on him after a very brief attempt to get him off safely. I'm glad that I'm not the sort of person who would indiscriminately drown a bunch of harmless spiders, but I'm also aware that it doesn't take a whole lot of incentive for me to put one under the faucet.
As far as comparing a dog and a mosquito, I think there are good, physical reasons why that isn't arbitrary. After all, we know with near-certainty when a dog is feeling pain, whereas I'm not even certain that mosquitos feel anything like pain, or even 'feel' anything at all. I like to err on the side of caution, but there's significantly more utility in helping dogs than mosquitos.
Cows and turkeys, though? I think you'd be surprised just how much preservation instinct they still have, despite the best breeding efforts of thousands of years of human intervention. And I wouldn't recommend trying to threaten a panda bear...
My interests are not in what the animals do, or how smart they are, or how much I like their appearance. Most of my interest lies in whether something I'm doing is causing them pain, or whether something I could be doing might help prevent or alleviate it.