Well you definitely wouldn't like Perilous Hearts then!

Try Rode to Rage or Technoshock with Locudus's sound mod (only animal is a robotic mosquito) for shooters, and run for your life, oh shit, Kringle Crash, Bloodshed Release the Pain, Scrolling Battles 9.6, The shooter concept demo, Crime Hunter, 2D Platformer, Deathmatch Project Alpha, or The Blood Rain for side scrollers.
Also try legacy as an adventure game, and maybe Sarah and the Castle of witchcraft and wizardry as well if you are comfortable with killing a giant magical snake monster (not graphic or actually required to beat the game) or making a cat sleep with a stunning spell.
Those are all the games I can think of that I'm sure don't have animal killing in them, that fit your genres.  You can grab most of those right from audiogames.net by looking under the games link or using the combo box on the front page, but you'll have to get The Blood Rain direct from ch-games.ru/en and Legacy from Right here
You can get the Technoshock mod from this link and you can get Shooter, Crime Hunter, 2D Platformer, and Scrolling Battles 9.6 (which is in the old games archive) from the "masonasons.me" website.

Super Liam also fits this bill, as there are dogs in the game but they're robot dogs.
Shades of Doom is a fairly primitive adventure/shooter hybrid, where all you really kill are demons.
Monkey Business? Anyone remind me if you actually kill animals in that game? It's been awhile.
I'd recommend A Hero's Call, as it's got great adventure potential (seriously, you will get lost), but there are a lot of animals in that game (wolves, bats, ravens) and if you opt not to kill them, it may be difficult to gain levels.

Thanks for the suggestions everyone, much appreciated. Actually, I have tried a blind legend. There is animal killing in that one.

Don't be a jerk Zarvox, you know very well that STW has graphic animal killing in it, not cool man.

And yeah, ABL has a bear in that cave, that large bird that attacks you on the mountain after the Marauder, and that wild bore in the forest path to the demon, among others probably.
I wouldn't suggest Shades of Doom because you kill birds, dogs, and rats in that one.
AHC is also definitely out, not really sure why it or SOD were even mentioned at all to be honest...
I'm quite confident that GMA Tank Commander has no animal killing in it though, and I suppose that would be considered a shooter so maybe you should check it out.

How about Alien Outback?

Super Liam also has rats, and berds, and snakes.

rats in the sewers, birds as the ambiance, and snakes in the woods

Rats in the sewer, and you do kill those; admittedly they don't sound much like rats.
Bears in the woods, and snakes. You are supposed to jump the snakes, but if you step on them, you get hurt, so maybe that's okay? Bears definitely attack though.

I apologize. I forgot this when suggesting the game as viable. That wasn't an intent to troll. It was me forgetting that those things were bears and rats.

What about the gate? D