Is there a biological mechanism for evolution encoded into our DNA?
This entire answer will be long, so read the short part first, then read the rest if you (or anyone else) is curious. Citations are included in the long section. I can include additional citations in the short section if needed.

Long Story Short

Your question touches on some common misconceptions about how the evolutionary process. Organisms don't "want" to evolve traits. Traits evolve through the biological processes of random mutation and natural selection. 

Organisms do not "want" to evolve traits.  (Well, OK, I'd love to evolve an extra pair of hands but that is not possible.) Natural selection works by modifying existing traits. Your turtle can stare all she wants at food out of reach but she will not evolve a longer neck. Instead, natural variation exists among neck lengths of the turtles because of variation of the genes that determine features related to overall boxy size. Those individuals with longer necks may be able to get a bit more food, live a little longer, and reproduce a little more. They will pass along their genes to their offspring, so perhaps more of their offspring will also have longer necks. Over many generations, the turtles may have somewhat longer necks. 

A common misconception is that the traits of organisms are precisely adapted for a specific need. They are not, for a few reasons. First, natural selection occurs relative to the current environment. Adaptations that work well in one environment may not be so useful in another environment. Environments are rarely stable over evolutionary time so traits are subject to constant change. 

Next, as mentioned above, natural selection can only work on what traits are present. While an extra set of arms would be handy, I am a tetrapod. My four appendages, along with the appendages of all other tetrapods, trace back to our common ancestor. The appendages of all tetrapods are modifications of that ancestral trait. 

Finally, organisms haven't "sampled" the entire realm of possible mutations and combinations of mutations. In other words, a certain mutation or set of mutations might actually be able to adaptively improve a particular trait in the current environment but, if the mutations never occur, then the improvement can never happen.

We only need to look at ourselves to realize how imperfectly adapted we are. We get bad backs and knees because our bodies weren't designed to walk upright. We evolved from quadrupedal organisms. This has happened so recently that changes in the structure of our knees and backs haven't yet evolved (and may never).  Search the internet for the "blind spot" eye test. We have a mass of blood vessels in front of the retina of our eyes, which reduces our visual accuity. We often have to have teeth pulled from our jaws because the flattening of our face (relative to our australopithicine ancestors) has shorted our jaws. We don't have as much room for our teeth but we have not evolved a reduced number of teeth.

As for human technology being able to make direct changes to our DNA to improve our adaptability, I would say no. While I do not have the ability to see into the future, the complexity of our genome, and more specifically how genes are regulated, suggests to me this would be a very daunting if not impossible task.  See the long answer below for more on regulatory genes but the gist is that a small set of regulatory genes control most of the other genes (including other regulatory genes). The interactions are extremely complex and we have a detailed understanding of very few of these interactions.  I speculate that affecting one such gene in a "positive" way is very likely to have many unintended negative consequences. 

Below are some simple math and other ideas to show you how mutations can lead to the many adaptive traits that you see among the diversity of life on earth.

Long Story


  how can so many specific (advanced) evolutionary leaps be otherwise explained?


Mutations occur at random throughout the genome. Most mutations will be neutral. That is, they are neither bad or good from an evolutionary viewpoint. The mutations are neutral because the genome for most organisms is non-functional. Mutations that occur in the functional regions of DNA (i.e., protein-encoding and related regions) are more likely to be detrimental (bad) because the mutation may negatively affect the function of the protein or even the ability to produce the protein.  However, some mutations are beneficial. The mutation may actually enhance the functionality of the protein or even produce new proteins.  

A couple of factors have to be considered regarding mutations.  The mutation rate is very low.  For example, Kumar and Subramanian (2002) compared the DNA sequences of 5669 protein-encoding genes from 326 species of mammals.  Their results suggested that the average mutation rate among mammals is 2.2 x 10$^{-9}$ per base pair (bp) per year.  This means that, on average, a point mutation has changed each DNA nucleotide position in the mammalian genome slightly more than twice (2.2 times) every billion (10$^9$) years. That's a lot of time!

However, this same rate occurs in every individual in the population, so you have to consider the population sizes of the organisms. So, let's do a simple exercise. Consider a species like the rock pocket mouse or another small mammal that has a very short generation time. For this simple example, let's assume the generation time is one year. That means that the mutation rate of 2.2 x 10$^{-9}$ per bp per year would then correspond to 2.2 x 10$^{-9}$ mutations per bp per generation. Generation time is important because new mutations are inherited only through reproduction.

Assume the average mammalian diploid genome is about 6 billion (6 x 10$^9$) nucleotides in size. The number of heritable mutations that occur in a single offspring is

$$(6 \times 10^9) \times (2.2 \times 10^{-9}) = 13.2.$$

Next, assume that about 2.5% of the mammalian genome is composed of functional, transcribed sequences that may affect the phenotype (the traits of the organism). That means that, of all the mutations that occur in every offspring every generation, about 2.5% could potentially affect the phenotype.  That is,

$$13.2 \times 0.025 = 0.33.$$

Still a small number.  But, now we have to account for population size. Small mammals, like mice and voles, generally have large population sizes.  Assume that the population of rock pocket mice contains 100,000 reproducing individuals. If so, then

$$0.33 \times 100,000 = 33,000,$$ 

which is the number of new heritable mutations that could occur in the population. Most of these mutations will be detrimental and removed from the population by natural selection but, if even a small fraction of these new mutations are beneficial, then natural selection can cause these beneficial mutations to increase rapidly in frequency in the population during future generations.

In humans, Nachman and Crowell (2000) estimated that the average mutation rate was 2.5 x 10$^{-8}$ mutations per bp per generation (not year), by comparing the genomes of humans and chimps. If we assume the same genome size and effective human population size of 500,000 individuals, then applying the same math suggests that 1,875,000 new mutations that potentially affect phenotype occur in the human population every generation. Again, only some of these will be beneficial but that is still the possibility of a number of new beneficial mutations. In evolutionary terms, a mouse or human generation is the blink of an eye.

How long would it take for a beneficial mutation to spread through a population? That depends on two things. How beneficial is the mutation (called the strength of selection, s) and the population size? To estimate how long it would take for a beneficial mutation to spread through a population, we can use the formula,

$$t = \frac{2}{s}\mathrm{ln}(2N_e),$$

where $t$ is time in generations, $s$ is the strength of selection, and $N_e$ is the effective population size (number of reproducing individuals). For the strength of selection, let's assume $s=0.01$, which is weak but positive natural selection. Going back to our rock pocket mice with $N_e = 100,000$, then the beneficial mutation would be spread throughout the population in only 2441 generations (remember, we're talking evolutionary time so 2000 years is nothing). If $N_e = 10,000$, the mutation spreads in only 1981 generations.  If we increase the strength of selection t 0.2, then the times are 122 and 99 years for population sizes of 100,000 and 10,000 years, respectively.

These "back of the napkin" calculations show just how quickly even weakly beneficial mutations can appear and spread throughout a population. Yet, this doesn't include other types of mutations like gene duplications that can also allow new proteins to evolve. For example, human ability to see red colors is due to a simple gene duplication (Nathans et al. 1996 and references therein). This duplication also explains the common form of red-green colorblindness.

Whew!

There's yet more to our mutational story. Consider humans and chimps, which are nearly identical from a genetic standpoint (between 96-99% depending on how you calculate it) yet they appear very different.  If humans and chimps diverged from their common ancestor within the past five million years, how could they differ so much? This question was initially posted by [King and Wilson (1975)]. They argued that mutations to structural proteins (like those that compose bones and muscles) would not be enough to explain the phenotype differences between humans and chimps. The proposed that regulatory genes are the key to understanding the big differences. Regulatory genes are those that control other genes, by turning them on or off and other important functions. Changes to the regulatory genes can cause fairly rapid changes to the phenotype.  

This understanding has led to the broad (and fascinating) field of evolutionary developmental biology. This field focuses on how mutations in regulatory genes associated with development (from embryo to adult) have had a long-term evolutionary impact. The field is rich with examples, but one cool one is associated with duck feet and bat wings. Let's begin with the embryo. Most vertebrate embryos have membranes between the digits (fingers and toes) during an early stage of development. For most vertebrates, the membranes are lost later in development. The small flaps of skin you have between your fingers are the remnants of your embryonic membranes. 

A set of regulatory genes called BMPs (and a couple of others) are responsible for causing the loss of the membrane in vertebrates. However, through different sets of mutations, the BMPs are not able to function in duck feet and bat hands. Thus, they both end up with membranes between their digits (Weatherbee et al. 2006).  Thus, two different mutations block the same set of developmental genes, leading to novel adaptations in two very different types of vertebrates. One final example is the evolution of bird feathers from scales. As you may know, birds are evolved from dinosaurs. It turns out that bird feathers and alligator scales (alligators are birds closest living relative) use the same regulatory genes to develop. The genes are BMP2 and SHH (sonic hedgehog for fans of the old computer game) (Harris et al. 2002). Other regulatory genes underlie the different types of feathers, like downy feathers and flight feathers (Harris et al. 2002).

Literature Cited

Harris, M.P. et al. 2002. Shh-Bmp2 Signaling module and the evolutionary origin and diversification of feathers. Journal of Experimental Biology 294: 160-178.

King, M.-C. and A.C. Wilson. 1975. Evolution at two levels in humans and chimpanzees. Science 188: 107-116.

Kumar, S. and S. Subramanian. 2002. Mutation rates in mammalian genomes. Proceedings of the National Academy of Sciences USA 99: 803-808.

Nachman, M.W. and S.L. Crowell. 2000. Estimate fo the mutation rate per nucleotide in humans. Genetics 156: 297-304.

Weatherbee, S.D. et al. 2006. Interdigital webbing retention in bat wings illustrates genetic changes underlying amniote limb diversification. Proceedings of the National Academy of Sciences USA 103: 15103-15107,