No question dogs are an ongoing resource comittment: time, money, attention, etcetera. Like all family members, it's always something.

Could a well-trained dog help avoid violating the Rule of Threes? I suppose a dog might be able to haul or assist you out of water or a smoke / particulat cloud to enable you to safely breathe no longer than three minutes after such unfortunate encounters.

A dog's body and body heat might make enough shelter to avoid death in three hours in a hypothermia-inducing environment. More than one dog might make a favorable outcome in such situations more certain. Not sure how this would work where the challenge was a hyperthermia-inducing environment.

Dogs may be trained to help you follow the scent of water, I suppose. And they could help dig to a water source if so trained. This could enhance the likelihood of finding water within three days. Of course, they form competitors for any water found.

Hunting could be enhanced by dog use. Perhaps they could be trained to hunt on their own and bring the prey to you. I hesitate to mention that if you find yourself three weeks without food, dog might be find its way on to a survivalist menu [also freeing its components for other survival uses].

For those who wish to avoid testing the potential lethality of three months without human contact, it is not unreasonable to think a dog could substitute to at least extend endurance to loneliness beyond three months.

Dog potential against my current survival categories of First Aid, Shelter, Fire, Water, Food, Navigation, Light, Signaling, Self-protection, Hygiene, and Morale?

Shelter, Water, and Food have been brielfly considered as Rule of Threes categories. I do not see how dogs could assist with Light, Signalling, or Hygiene [except as water plays a role in hygiene]. I suppose dogs could aid an injured person to move and so play a First Aid role.

Dogs could aid in self-rescue type navigation in finding the scent of other humans, their byways, vehicles, livestock, and habitations. Self-protection and morale do seem naturals for dogs.