Notice how I said almost any animals. Giant worms have no brains and animals like deer and rabbits that are skittish by nature would be useless anyways, but a legendary animal trainer should be able to train them anyways, because he's legendary. I mainly meant that predatory animals, like lions, wolverines, and crocodiles should be trainable. In real life, once large animals taste human flesh it isn't unheard of for them to start attacking lone women and children, and that's without being trained.
Well, a majority of animals (individuals, certainly, and probably also species) have intelligences and/or temperaments unsuited to war training. The predators and some vicious herbivores? Sure, although that category includes critters that no one in their right mind would train for war. Most animals? No.
Again, temperament is a huge obstacle. It's easier to get hippies to kill something than to get rabbits to kill something.
So why is, again, 'easier' defined as a switch and a block instead of a skill and time-based chance of success?
It is easier to train some animals to kill than others. That does not exclude the possibility of the vorpal bunny.
Barring extensive modification to the base psychology, it kinda does.
I can see three or four groups being created--animals easy to train for war, animals impossible to train for war, and a group or two of animals difficult to train for war--and perhaps a lengthy process allowing a population to become easier to war-train, but there should always be animals which you can't train for war.