How would an animal ever accumulate enough merit to warrant rebirth as a human?

As @hellyale suggests in his answer, the Jataka is a good source for tales about the lives of animals, their actions and its consequences.

Usually an animal attains merit by selfless acts of devotion - whether it is a dog running into a fire to save a child, or a cow fighting off a much larger animal to protect its calves, or a tiger not killing for wanton pleasure and not deriving pleasure in torturing its victims, such acts lead to an accumulation of merits.

Another way, though less certain, is, if one of the progeny from a previous life does great merit as a human or deva and offers it to her ancestors, in whatever realm they may be.

Yet another way: In the story of Manduka (Sanskrit: frog) Devaputta (Sanskrit: son of devas) (Chronicle of the Buddhas, Page 1123) - a frog happens to die while listening to the Buddha's sermon, and attains the Tavatimsa Deva realm. The story says the animals can't understand the content of the speech of the Buddha, but the vibrations of love and peace and wisdom can certainly enter the animal. Just as a stray dog understands a boy intends harm even before he picks up a stone to throw at it, some animals are attracted to peaceful people, and the dharma without the need for any words.

seems like a game of snakes and ladders in which - other than when playing as a human - its going to be all snakes.

Not really, the deva realms are not ideal only for those who are unenlightened, since they can get lost in the luxuries and pleasures.

However, when an Anagami dies (s)he heads to the deva or bramha realms to continue practicing until arhatship because it is better for practice at that stage - without the mundane worry of finding food and shelter or falling sick. The luxuries of that world won't affect their practice at all, but will only aid them.

Excellent line of inquiry.

Just as you have proved, it takes a long time, sometimes thousands and millions of rebirths as a lower life forms to accumulate said merit.

On the other hand, demigods can ruin or greatly increase their stores of merit (because their power is that much higher). A demigod can sink direectly into the deepest states of hell. Some demigods die even if they miss one meal.

Human incarnation is in the middle and has all the advantages the lower and higher incarnations do not have.

Everything I have said here is in the suttas and all Buddhist texts agree.

An important thing to mention regarding kamma and rebirth is that there are also past kamma that has not yet ripened and come to fruition. If the right conditions are present past kamma can come to fruition thereby resulting in a higher rebirth, e.g. as a human being.

It is like a seed. If one nurtures the seed with water, nutritious soil, sunlight and other conditions the seed might blossom and turn into a flower.

Kamma is fourfold and is divided into sections, i.e. "by way of function, order of ripening, time of ripening and place of ripening".

Kamma by time of ripening is further subdivided into 4 types according to the time of taking effect:

(i) Immediately effective (diμμhadhammavedan2ya) kamma is kamma which, if it is to ripen, must yield its results in the same existence in which it is performed; otherwise, if it does not meet the opportunity to ripen in the same existence, it becomes defunct. According to the Abhidhamma, of the seven javanas in a javana process, the first javana moment, being the weakest of all, generates immediately effective kamma.

(ii) Subsequently effective (upapajjavedan2ya) kamma is kamma which, if it is to ripen, must yield its results in the existence immediately fol- lowing that in which it is performed; otherwise it becomes defunct. This type of kamma is generated by the last javana moment in a javana proc- ess, which is the second weakest in the series.

(iii) Indefinitely effective (apar±pariyavedan2ya) kamma is kamma which can ripen at any time from the second future existence onwards, whenever it gains an opportunity to produce results. This kamma, gen- erated by the five intermediate javana moments of a cognitive process, never becomes defunct so long as the round of rebirths continues. No one, not even a Buddha or an Arahant, is exempt from experiencing the results of indefinitely effective kamma.

(iv) Defunct (ahosi) kamma: This term does not designate a special class of kamma, but applies to kamma that was due to ripen in either the present existence or the next existence but did not meet conditions con- ducive to its maturation. In the case of Arahants, all their accumulated kamma from the past which was due to ripen in future lives becomes defunct with their final passing away.

-- "Comprehensive Manual of Abhidhamma" by Ven. Bhikkhu Bodhi, p. 200-220.

As the effects of kamma is impermanent the dog would sooner or later be reborn in either lower, same or higher realms. Here the different types of kamma can come to fruition if the right conditions for them to ripen is met.

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.