Thanks for playing! Glad you thought it was cute!
There really isn't much more depth to it than what you described. The pet health increases when they have food and water, and you can put them to work to make money. you find more coins and up to 3 pets in the dungeons. When a pet is tired they earn less money. The faces of the pets change based on how much health they have and if they are awake, tired, or asleep. They lose health when they aren't cleaned up after or begin to starve. The more dungeons you clear the harder the puzzles get. With all three pets, management comes more into play, but you earn 3 times the money. Also after the work counter hits 99 and resets back to 0 that pet earns twice the income. this can happen twice for each pet.