Sometimes he will, sometimes he wont. I cant seem to figure out the conditions, but during the game there will be no more cards I can play, I will have 2 or more mana left, and the bot ends the turn instead of using the hero power. I have seen it use the hero power before, thus this seems sporadic.
Also, hunter would never attack with King Crush charge. He would just summon him and then end the turn :/ Update: Seems that bot would not use not only King Crush charge, but also charges that are granted by rhino.
Bot does not also use hero power when it's available with Warrior and Mage. Probably does this with all classes.