The key here is in the third word of your question - DNA is LIKE a code. It is not a code. That's just a useful analogy, but like any analogy, it has its limitations. When you get down to its most basic, DNA is entirely mechanical and chemical. Enzymes don't "read" DNA, they just carry out chemical reactions. Transcription factors don't "recognize" sequences of DNA, they simply bind due to chemical affinity. Granted, all of this is at an extremely high level of complexity, but it's still chemistry.
I think you're overestimating the amount of randomness that you'd see, though. It's still all bound by the laws of chemistry, which means that there's an inherent order to it. Granted, the sequence of nucleotides would be fairly random, but we're talking about extremely long time frames, and extremely simple chunks of RNA at first (somewhere around 100 nucleotides, probably), so that's not really an issue.
I'd argue that the idea of a designer isn't unscientific because there's no direct evidence, though. That would mean that proposing a designer would simply be jumping to an unjustified conclusion... not necessarily unscientific in and of itself. The idea of a designer is unscientific because it's not testable. If you're talking about unknown designers with god-like abilities and power (or in the case of the Intelligent Design and Creationist movements, the protestant version of the Christian God), then your run into problems. There's literally no piece of evidence that could ever be discovered that could *disprove* design. Something looks designed? It was designed. Something looks like it evolved naturally? It was designed to *look* like it evolved naturally. Since it's untestable, there's no way to gauge it's accuracy, and it's useless as an explanation.
As usual, though, CRR is lying about evolution. Aside from grossly misrepresenting "genetic entropy" and using an intentionally-wrong version of evolution, he's putting far too much stock in the ENCODE project (which used methods and analyses that were problematic, at best).