Is it reasonable to think of it like this: RNA is procedural, and DNA is declarative? RNA : PostScript :: DNA : PDF?

Actually when writing this comment, I was thinking about the LISP paradigm: RNA is both code and data.

except so much messier, it's more like code and data in the sense that self-modifying machine code is.

Not really. The 3D structure of DNA is also important for its function. The difference is that RNA is more reactive (chemically speaking), else they are pretty much equivalent.

A better way to think of it I think is that RNA is both software and hardware.

Yes but RNA is much more destructible, whereas DNA is more resilient. RNAses(enzymes that degrade RNA) are everywhere whereas DNA is double stranded and the conditions required for its degradation are harsher. DNA may have 'evolved' as an aberrant form of RNA and because of its durability, it has an increased lifetime. This renders it more apt to serve as an archive that is copied multiple times. An organism relying only on RNA for information storage would have to deal with the shorter lifetime of RNA and that it is more damage prone. That organism would have to spend more energy just propagating its blueprint faithfully and so its population would not grow as fast. Still DNA is not so durable that its sequence is not malleable, like proteins. If I were designing the most robust organism, I would prefer to encode information in amino acids. I would then design a polymerase that would read the protein and copy it. The problem is that proteins are so robust that they are not easily unfolded to be read.

Perhaps a tenuous analogy can be made to information storage in transistors versus magnetic media. Transistors can serve as storage medium as well as logic gates. Yet power fluctuations can corrupt their function. Magnetic storage however can be read many times and is more durable with respect to environmental insults. Of course the most durable form of storage might be laser etching defects in a diamond disc but at the cost of a slow energy intensive process.

How accurate is it to say that life is essentially the result of a biological computing process? That we are quite literally built out of tiny biological machines?

Depends how you define it. A biological system is not deterministic (at least on the scale we typically observe) so from that point no. Else you can see a cell or an organism as one computer but then you can also define the universe one big computer.

I would rather say life is the result of an optimization process.

Isn't it an unsolved problem whether or not physical reality is deterministic?

At minimum it's true in the same sense that physics is a computable process, though that probably isn't very insightful.

but the amazing coincidence is that DNA strongly resembles a binary code. There are only two kinds of base pairs- AT and CG - In itself this is a striking similarity to the way we use transistors to define a 0 or a 1.The pairwise complementarity serves as an error correcting algorithm. I am rather surprised though that no one has constructed a biological model of the cyclical redundancy check so common in computer hardware. Error correction in DNA is haphazard in the sense that when DNA polymerase detects a mismatch of a base pair, it really doe not 'know' which one to correct. Errors have a 50% chance of being corrected the 'wrong' way. Still that's better than no correction because broken DNA always fails. A CRC check at every 1000 base pairs would virtually eliminate genetic mutation and all the diseases associated with DNA degradation (cancer, aging, etc)

But that wouldn't necessarily be ideal long term, because there would be near zero adaptability on account of advantageous mutations. Unless the organism could adapt is behavior, it would be really vulnerable to environmental disruption.