How would an animal ever accumulate enough merit to warrant rebirth as a human?
In the Jataka tales (Similar to Aseop's Fables ) there are many stories about the Buddha's various past lives. These are tales for children and normally come with a moral. 

There are a ton of them, and they normally end with the animal the Buddha is reborn as helping out in someway that is aligned with the teachings. 

You can see a few of them summarized quickly without the entire back plot of a fable here.

It is absolutely correct that in the Buddhist worldview a human rebirth is of great advantage. Animal existence is not what most of us would consider fun. As one teacher once reminded me, you might be a pet, or you might be reborn a deer in a forest of wolves. An animal life is mainly fear and instinct. The characteristic of their realm is that of Ignorance. 

As you yourself pointed out, your dog (Who is lucky and has food, shelter, and someone looking out for them) is not often reflective. 

Anger the characteristic of the hell realms is more damaging, and can lead to a large snake on the game of chutes and ladders. 

Your dog though, in your ownership, is likely not accumulating massive amounts of negative karma. 

I hope this helped to answer your question.