Maybe I just haven't noticed, but it my pet never seems to take damage. Can he be killed? What happens if he can, and does die?
No, your pet cannot die. If your pet gets too low on life, they'll panic, run away and return when they recover. They'll regenerate life slowly while fleeing, so if you wait a bit, they'll come back on their own.
You can speed the healing process by giving your pet a health potion. I believe this requires that you shift-click or shift-hotkey (ie, if your health potion is 1, press Shift-1) a health potion.
You can also make your pet tougher by feeding it fish, so they don't take as much damage, and therefore don't run away as easily. Different fish have different attributes and last for different amount of time, so experiment to find the one that works best for you, and then feed it that fish before/during a fight with a large boss or other dangerous area of the game.