As usual your assumptions are completely wrong, electric fences are never continuously ON, both because it would be illegal, and (more importantly) it would greatly reduce battery life. Electric fences are designed to stop cows, horses etc. - not tiny animals, I'm not surprised one would kill small animals, however their normal usage would mean that it's pretty well impossible for them to affect small animals.
A simple physical fence would keep the dog out, but you're not going to stop cats, not even with an electric fence.