Junk DNA and "random" mutations
pure randomness does not seem like something that can survive in a long evolutionary process.


I think this is because you don't quite understand (no offense) the interplay between mutations and natural selection. You should have a look at Understanding Evolution.


  The overall discussion of mutagenesis I can find focuses entirely on "damage"


This is because the vast majority of mutations are deleterious (=damaging). But obviously, not all mutations are deleterious, otherwise no adaptation would be possible (only purifying selection would occur).


  In a similar vein, I don't see not much of anything about evolution and [..]


I don't understand this paragraph.


  Basically all of evolution and natural selection seems about how to survive into fertility and select a partner [..]


No! There is much more to evolution than that. Note that the semantic of evolution and natural selection is wrong as they are used in your sentence. Again, check out an introductory course.


  I think that this kind of viewing the message in the DNA as isolated from its medium


No! The environment is a fundamental tenet in evolutionary biology. The association between genotypes to fitness is a function of the environment as well.

Are mutations random?

The mutation rate varies dramatically among species and along genomes (roughly speaking from $10^{-2}$ to $10^{-10}$). For example microsatellites (repeated sequences) are highly mutable. Note also that the probability of a given nucleotide to mutate depends also on the exact nucleotides that are around it (Rob Ness; Personal communication). The mutation rate is also a plastic trait (and is therefore under selection) in some species (of plants typically). The environment affect the mutation rate and the environment can be chosen by some species (avoidance) or even constructed in some species (niche construction). In multicellular organisms, the age of the father correlates with the number of new transmitted mutations as well. However I have never heard of any studies showing such behaviours in order to affect the mutation rate. Epigenetic changes (does not concern all species) can also the mutation rate and even into what new nucleotide the old nucleotide will be mutated into (with a given probability). Mutations are more likely to be transitions than transversions. If a specific region is under environment-depend selection, then a mutation that would allow to drastically increase the mutation rate at this nucleotide whenever the organism has a low fitness (or even whenever the trait of the organism is unfit for the specific environment) could reach fixation. This process (and maybe others I am not aware of) are I think encapsulated within a hypothesis called (and it is a horrible name as it can yield to confusion) adaptive mutation hypothesis. But really don't start studying any field of science by jumping into an advanced subject (or even worst, a controversial subject). You should start with an introduction.

However, the impact on fitness (and on phenotype) of a new mutation is essentially random. An organism has absolutely no way to know what protein will be caused by a given new mutation. Selection then select upon the variance created by the mutations, the variance decreases and adaptation occurs. The impact of new mutations seem to follow more or less two weighted gamma distributions (added to which a bump of probability density for lethal mutations).