Rorschach butchers two German Shepherds following a mental breakdown. We don't see the act itself, but we do see the dog's bloody corpses.
Later on Hollis Mason's dog is implied to be physically assaulted or killed, but it isn't explicit.
nothing directly shown or graphic, but there were many animals (frogs, birds) in Veidt's greenhouse area that he let snow into, which would have 100% killed them.
When Dr Manhattan takes Laurie to Mars, there is a brief scene where she struggles to breathe due to the lack of atmosphere on the planet. Manhattan quickly fixes that however.
Rorshach is incredibly violent, morally absolute and paranoid.
At one point, an (inept) counselor is sent in to diagnose him.
However, there are no diagnosed mentally-ill violent characters.
Later on Hollis Mason's dog is implied to be physically assaulted or killed, but it isn't explicit.