"Interestingly, human reincarnation is not taught, but what of the rest of living things?"
THe Bible really doesn't say much about the rest of living things. I presume you mean flora and fauna?
Man is the only creature that is a created in the image of God: Body, soul, and spirit. Animals are referred to in the Bible as creatures for the benefit/stewardship of man, but nothing is taught about their souls. God does know when a sparrow falls, so I know He cares for them. (I always like to think that my dog, Peaches, will be in heaven, but the Bible is pretty silent about that, as far as I know.)
As for plants? The seed falls to the ground, voila! ;o)
Would it be Heaven for you, if you could never be with Peaches again?