By default your sled dogs can be killed, but an in-game options setting will make them always survive.
Reference: https://www.polygon.com/2020/10/22/21528967/dog-survival-game-red-lantern-death-toggle
Your dogs will be sad if you offer them no food when you choose the option to feed them. However, it is quickly remedied by feeding them, and the more food you give them, the happier they get.
Reference: https://www.polygon.com/2020/10/22/21528967/dog-survival-game-red-lantern-death-toggle