We don't put adverbs between certain words because those who taught us language (whether through formal instruction or natural methods) don't put adverbs between certain words. We do, however, sometimes violate these rules; the evidence is in the way all languages change and evolve.
A native speaker's understanding of grammar is unconscious, yes, but it has still been learned. If you drive a car, especially if you have been doing so for a long time, you do a lot of it instinctively, and you don't usually have to think very hard; however, you still had to learn to drive. (It's not a perfect analogy, of course, since when we learn to drive we do so by formal instruction first and then our skills are fine-tuned by experience, but it still illustrates the point.)
These syntactic rules are certainly not universal. In modern German declarative sentences, for example, the inflected verb is usually the second item in a main clause (and if the first item is not the subject, the subject follows the inflected verb), and one of the last items in a subordinate clause. A German doesn't have to think about these rules at all (except when the sentence becomes very long and constituent parts of a verb phrase end up a great distance from one another), but the difficulty most non-native speakers have with such structures indicates that this "innate" understanding of the underlying grammar is peculiar to German speakers. However, we know on other grounds that your biological heritage has no effect whatsoever on your ability to learn a specific language.
What you're basically saying is that we humans have the ability to formulate highly complex and subtle rules even though they are unable to explain them, and without being conscious of the process. But that doesn't mean that these rules do not have to be learnt.
This is where we get carried away by terminology if we are not careful. The moment we hear the word "unconscious", we understand "instinctive", and when we hear "instinctive", we understand "innate" and therefore "doesn't have to be learned".
And if all languages do have superficial similarities -- all languages having verbs, for example -- that doesn't mean much. Studying sufferers of various types of aphasia, for example, we know that different parts of the brain are responsible for certain parts of language, but that is about as far as it goes. In fact, different languages have very different ways of expressing the same thing, and different ways of using verbs. In European languages we inflect our verbs to indicate person, time, tense and so on; non-European languages may use completely different mechanisms, such as particles or honorifics. The only real similarity is that we all have words for describing actions, but how else are you going to develop a language? Besides, if our languages, as seems likely, are all related to one another, it's hardly a surprise that they have superficial similarities.