Just to add to this string of thought and create more thought hodgepodge...
Immortality and Mortality can be explained through evolution if we examine the cost/benefits in reproductive efforts. It`s a little known fact that Immortal organism actually do exist in nature (usually plants). I admit this does not follow your definition of "absolutely unkillable", but such organisms could conceivably live forever. Here are characteristics of both immortality and mortality in nature.
-Mortality keeps a species adaptive through recombination of genes passed to the offspring. Certain heritable traits will have advantages over others which are then selected by nature. Through death, new adaptations arise (and gradually, evolution). So from an entirely evolutionary standpoint, for a species, mortality seems to be a very good strategy.
-more energy needs to devoted to reproduction, and less is needed for personal survival.
-Organism concentrate more on individual survival and less on reproductive potential. This strategy tries to maximize the number of offspring produced by staying alive itself, and producing progeny.
-Immortal organism will be able to weather "bad years". Say a plant faces an exceptionnally bad year, and wastes all it`s reproductive energy wasting seeds in unseasonably cold weather. All those progeny will be lost, and the results could be disastrous for the species. An immortal organism would tend survive this weather by concentrating on it`s own survival, and at the same time didn`t waste all its energy on reproducing.
There are always costs/benefits to behavior in nature... laws that we humans are also subject to.