Why linguistic form? Historical accidents, and communicative need.
Why linguistic structure? Modification (human language has a fundamental property of one concept modifying another-- by another name, Chomsky calls this "Merge"), and communicative need.
In short, I think it's important to separate the two.
...but my dissertation is still in progress.
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Edit: reading your blog entry--
First, why do you treat language as information/code? I thought it was established, at least in pragmatic theory, that language is not a code in a technical sense. I've always wondered about that. In some sense, of course it must convey information. But what exactly is it conveying? Are there 1s and 0s? Or are there just hints for the hearer to associated particular concepts that are somewhat loosely related to similar concepts in the mind of the speaker? If we're going to talk about information, then what exactly is information? And if there is information, why not just say it directly? Why do languages vary so much, if it's just "information"? Or is there more going on?
For example, if the phonological form of a word is much longer than its Shannon information, we could improve our efficiency by making that word shorter. Conversely, if the phonological form of a word is much shorter than its Shannon information, we could improve our efficiency by lengthening that word, thereby freeing up a short phonological form to be used by a different, more frequent word.
I suppose this makes sense, but I'm not sure it works exactly that way. For one thing, not all short word forms are utilized in languages. And some languages have about a dozen phonemes while others have toward 150. Language is more complex than just information: it's also form. Adding more contrasts to the signal would also be more efficient (encode more information), but then it would run the risk of being less efficient for the hearer. Probably in some way all of these factors balance out.
In the end I do wonder if the arbitrariness of language (and I mean that to extend to syntactic form, for reasons I'm working out in my dissertation) is part of how it works: we refer to culturally defined shortcuts ("lexical items") rather than some objectively 'best' code.
So in short, I'm still lost about what "information theory" has to do with language-- what information does language encode anyway? If we can answer that and get to a formalization, this will all start making sense, but there are still some complicated pieces. One is, as I said, that we refer to shared lexical entries, not to objective information. (We can quantify those references to the lexical entries as information, of course, but this gets complicated, because some of the work is done by the lexicon then.)
In a block code, we encode sequences of message characters instead of individual message characters.
Ooh, now I'm starting to wonder if this has to do with syntax, periphrasis and 'structure'

First, observe that a substantial portion of linguistic meaning is compositional
If you're referring to processing (which is what gets us historical change and then current usage), this is complicated. I believe there are shortcuts so that everything isn't strictly compositional. Much is, but not everything. In terms of quantifying any information in the signal then, it becomes tricky. One imbalance might be idioms where syntactic form is still relatively transparent despite lexicalization. And if non-compositionality (at least for processing, think "chunking") plays a part, then this also gets tricky. There's a balance between competence-compositionality and processing-chunking. Why does the signal evolve faster? Because we need to have some way to reconstruct some of the shortcuts in our lexicon, something like an index to search a database. But eventually some of those shortcuts leave the competence-compositionality behind ("grammaticalization" I suppose).
In other words, because natural language is compositional, it assumes that different components of a message are statistically independent! However, different parts of linguistic messages are highly dependent. Thus, natural language implicitly relies on a family of probability distributions that is far too constrained to capture the true distribution over messages for optimal source coding.

Much of parsing, and probably a substantial amount of speaking, involves just knowing what's coming next. This is one area where native speakers far surpass learners-- try speaking in a noisy environment (pun intended?) and just see how much you can still understand despite the weak signal.
This view suggests that language is far from optimal in the sense of information theory. It does not, however, mean that there is not any pressure in the direction of optimal behavior ...
While I'm not a favor of unfounded "everything balances out" arguments in general, this reminds me of the "simplification" problem in historical linguistics-- "simplification" can be used to explain many changes, and it seems natural. Yet the "complexity" (whatever that may refer to) must come from somewhere. So languages are not, on average, becoming simpler, or in the end they'd just stop changing and be, well, simple. So I imagine that this is the same here-- information theoretic concerns are a force acting on language, but just one of a number of such forces.
We could, for example, use the same block coding trick as before: instead of encoding grammatical elements as separate message components, encode ensembles of grammatical elements simultaneously. Now, however, we are dealing with a much larger space of message components, such as all the words in the language, so we'd want to get a lot of "bang for our buck:" the increase in fit of our low-dimensional approximation must be very large relative to the additional parameter introduced by the larger block.
This (like the going to / gonna example) is fundamental to 'Construction Grammar'. And I think you have a point. I like that it goes deeper than just observations of meaning, though. And I'm wishing I understood the math a bit better (it's me, not your explanations-- I haven't looked at equations like that for a long time, and you're reminding me I probably should). This is interesting (and see my comment above).
...the frequency effects are assumed to be given, as some kind of basic feature of how brains work.
This I can help you with. There's a very interesting (thesis and now) book:
Zeldes, A. (2012). Productivity in argument selection from morphology to syntax. Berlin: De Gruyter Mouton.
This framing suggests that the role of frequency effects in grammaticalization itself may reflect an adaptation towards some balance between incrementality and brevity.

I think this is right. I'm not sure whether it's blatantly obvious or especially insightful. I haven't seen it phrased that way before, anyway, so let's call it insightful. I'm a bit inspired by the post at least.
So while I'm trying to write the formal proposal for my dissertation, these thoughts are distracting me about what's next-- I'll be very interested to follow up on some of this after I have something more concrete from my research. I'm curious if it would tie in. I'm looking at the (cross-linguistic) data-driven side of things, but ending up in the same way intersecting with grammaticalization, construction grammar vs. generativism, and so forth.