We share 50% of our DNA with a cabbage. Most of this commonly shared DNA no longer codes for proteins in humans but in a cabbage obviously a lot more of that DNA does code for proteins otherwise there would be no cabbage.
To say that we expect coding DNA to have patterns but non-coding DNA not to have patterns is simply erroneous. The DNA which is currently non-coding in humans is inherited DNA from ancestors which used to code proteins, therefore we would expect to see "meaningful patterns" in both coding and non-coding DNA as non-coding DNA is merely a remnant of DNA that used to code. Just as you'd expect to see patterns of letters forming words in pages of a book that has been ripped up and no longer tells a complete story.
Would have to do a bit of checking up but I'm not sure this is quite correct. I'd be willing to bet that a fair bit of junk DNA is just junk that has always been junk. Given that replication is imperfect and that benign but useless DNA does not seem to be selected against (because lots of genomes have stacks of it) I can't see any reason why some of it couldn't have just accumulated without ever having had a useful function anywhere.
You could check junk DNA for protein coding as the codes for the common proteins are well known already. If you find stretches that don't contain these codes you may have found some "genuine junk that always was junk".
As for language it is possible to trace languages just like it is DNA. I don't recall the name of the study but I read about it in a book called "The Language of the Genes". It was possible to identify derivatives of languages such as accents, an extension of that being some regions using different/new words, all the way up to completely different languages. The study examined languages for similarities and by analysing the closest related and mapping them to geographic locations guess what it showed? It showed a pattern of human movement throughout the world over time, as if we had spread out to populate the planet, so it proves neither evolution or the Quran which both suggest human migration. What it does show however is that it is not the case that some God one day at the tower of Babel suddenly made us all speak different languages (or whatever it is you are suggesting - you didn't actually make a point.)
These languages influenced each other over time (such as in English speaking countries where they use the odd French word such as "Restaurant"). So overall one would expect languages to have similarities in them and also most common words such as
One, Happy, Sad, Sun, etc.
True for some languages but there are completely unrelated languages where this does not apply. However given that we all share the same basic brains it is quite likely that there is one fundamental "coding" underlying all human language. The actual words chosen wouldn't affect this as they are basically an arbitrary facade put on top of the basic coding.