I've heated with wood for four winters so far. My four cats love the wood stove and have never gotten hurt. During the cold season they claim their spots near it or underneath it (it's up on legs) but never get too close. They run away from the intense heat when I open the door to load more wood, then come back when they're sure I'm done. They learn quickly.
However, we did have a blind senile dog who was always running into things and bumped into the stove a couple times. So we made sure to place sturdy barriers around it for his safety when we weren't around to supervise. Hog panel surrounding the stove works, as long as it's secured so it doesn't get knocked over. A sturdier, but more expensive option would be those safety fences designed to keep babies away from the fireplace. Just be sure it's made of metal.
