Do the same rules of Karma apply to animals? Animals are sentient beings and therefore are also bound to the cycle of rebirth.

However, animals are far less intelligent than humans, and are certainly not smart enough to follow a particular code of conduct, or take decisions which are against their nature.

Does Karma and rebirth work differently for animals? Or are animals pretty much doomed to remain animals (or worse) because their nature, for instance, causes them to kill?
Karma is a universal law. It works the same for all beings. Whether you are a god, human or an animal, what matters is the strength of craving, aversion and ignorance. 

Yes, if you sink below the level of humans, it's pretty difficult to come back. That's why the hells are called the great houses of Sansara.