Nature is the combination of matter and consciousness. Consciousness is the essence of life, matter just a container for it. No doubt animals are conscious, and science has proven that plants also exhibit consciousness to some degree. But among all conscious entities, human beings are most rare: Human life is that moment in history when nature has the opportunity to awaken to the fact that it has a soul.
The Bhagavad Gita compares the combination of matter and consciousness to a fire covered by smoke, a mirror covered by dust, and an embryo covered by the womb. The fact that the consciousness of lower forms of life is more covered than that of human beings should not discourage you. Such is the influence of karmic rule. On the other hand, you should be encouraged by the fact that you have such a rare human birth and should take steps to realize the purpose of life and the extent to which you exist. Realizing this is the end of all fear and the beginning of real joy.
If karma determines the happiness and distress of a human being, what determines this for animals? Why does one animal live a long life while another is eaten shortly after its birth? What law is governing the life of an animal?
Animal life, according to Hindu scripture, is but a karmic reaction to the way humans act. Embodied souls pass through many species of life, but only as human beings do they accrue karma, as only in that condition is one's free will fully manifest.
With that free will comes responsibility. Humans who misuse their free will have to loose it, at least for a time. Chapter 16 of the Bhagavad Gita describes such human beings as having to take birth in lower species of life. So some souls in human dress later wear the dress of an animal, and the degree of suffering or enjoyment they have in animal life is directly related to what they did in their past human life.
Animals cannot create new karma because they are completely controlled by their nature. Similar to criminals incarcerated by the state, souls wearing animal bodies serve out their karmic sentences until they again are eligible for a human form of life, in which they will be given another chance to act responsibly and progress toward liberation.
What would a human being have to do to be reincarnated as, say, a chimpanzee--as opposed to being reincarnated again as a human or a higher form of being?
The general rule is that one's future birth is determined by desire. In accordance with one's desires, material nature gives the soul the body it deserves, a body that best facilitates the satisfaction of those desires. For example, the body of a hog facilitates gluttony much more than does the human body, and a chimpanzee in his natural habitat has the capability to enjoy unrestricted sexual activity far beyond that of a human being. Human life is best suited for spiritual pursuit; this is what sets it apart from other species. If in human dress one does not care for this opportunity and simply pursues that which another species is best suited for, material nature provides the soul the facility—the appropriate body—to satisfy those pursuits.
With regard to determining one's future destination, the Bhagavad Gita says that a predominance of tamas (ignorance or indolence) in one's psyche--manifesting in human life as the proclivity for over indulgence in sensual enjoyment--pushes a person downward in the next life. A predominance of rajas (concern for material improvement) in one's psyche keeps a person perpetually in the human condition, while the predominance of sattva (goodness or virtue) in one's psyche propels a person upward in the spiritual direction.
What determines how an animal is reborn after it dies?
The entire life of an animal, including its next birth, is determined by what that soul did in its previous human life. A person can act in human life such that his or her karma takes a number of births in lower species of life before returning to human life. As the Gita says, "mudha janmani janmani"--a person may have to take birth after birth in lower species of life if they spend their time in human life too foolishly.
The Bhagavad Gita teaches that it is nearly impossible to sort out the details of one's karmic entanglement, as there are so many factors involved. It compares the soul’s constant changing of bodies to a person discarding useless clothing and putting on new ones. Each change of body is just another cloak of karma appearing as physical and emotional characteristics, such as parents, nationality, spouse, children, friends, fortune, fame, health, and finally the time and circumstances of one’s death.