Is symmetry in animals due to DNA or natural selection? By symmetry, I mean the left side is an almost (not perfect) reflection of the right half. Is this due to DNA trying to balance the two half or is it due to natural selection where unbalanced organisms die off more frequently and thus reproduces less. Or is it due to some other factor I had not considered?
Before you consider the development of bilatteral symmetry (which is what you are talking about), you need to look at the overall changes relating to animal body plans.
To begin with, sponges (Porifera) completely lack symmetry and came very early in the evolution of animals. Basically, it's a large cluster of growing cells with some differentiation (ie: choanocytes, amoebocytes, etc.). So the lack of of symmetry is simply related to the lack of complexity.
Then, radial symmetry, which has a "top" and "bottom" but no "left/right" came into play with phyla like Cnidaria. An evolutionary jump from assymetry to bilateral, would be astronomical, so we can think of radial symmetry as an intermediate relating to the increasing complexity of body plan development.
On Bilatteral Symmetry, one key thing is the cephalization. Cephalization is where a majority of the nervous tissue and sensory tissue are clustered in one area (a head). In the evolution of animals, Cephalization came with the evolution of Flatworms (Ptalyhelminthes) where sensory tissue such as ocelli (eye spots) were placed in the head. With the head developed, a "sense of direction" can develop where the right/left/dorsal/ventral sides of the body naturally become distinguished.