I wouldn't knowingly eat dog. I especially wouldn't knowingly eat cat. (I've never had a dog in my house, but I've had a bunch of cats as pets over the years.) Yes, it's cultural. I am an omnivore, but I do draw my personal line at certain animals. Dogs, cats, primates, endangered species (I stopped eating bluefin tuna something like 15 years ago.)
That being said, I grew up in an adventurous household, and have eaten a wide variety of animals: rattlesnake, alligator, kangaroo, bear, rabbit, squirrel (this one is REALLY good).