This is an extremely dismissive comment, plus you are way off basis that it is underdeveloped. It is just new/underdeveloped for OP.
Literal books have been written about this exact idea by Zellig Harris. The reason it is not more well known is that Chomsky (his student) became famous and reinterpreted a couple of his ideas (using mathematical logic for the basis of his work rather than Harris's set-theoretic base). But while Chomsky had to keep creating castles in the sky (including the idea of universal grammar/external metalanguage) to patch the holes in his theory Harris's theory needs none of that and is self-organizing and self-sufficient.
An alternate (and consistent with OPs idea) list of layers of processing a language.
1. Phonemic distinctions.
2. Words and morphemes, with their main meanings. (The word with its morphemes is an umbrella that also encompasses gesture, visual, and any other perceptual information related to the word. The words we use are not separate entities from our experiences. In the same way, mirror neurons aren't a thing. We mirror because we only have one set of machinery, and so to understand the other person is literally to hallucinate experiencing it to some degree.)
3. Word dependencies (the argument requirement of each word).
4. The selection of each word (the dependencies that have greater than average likelihood).
5. The canonical or preferred linearization (word order) and its alternatives.
6. The main reductions (variant shapes of words), their domain (a particular word or all words in a position) and the conditions for applying them.
Linearization is one of the last steps of producing a sentence. According to Operator grammar theory, most language complications come from the simplifications we produce as shortcuts (such as pronouns, contractions, conjunctions, etc).
So your first two questions are just the memory of the words/ideas/objects/etc combined with an operator and our perceptions in context do the choosing. Very much like an AST, but statistical (for word likeliness), as opposed to deterministic.
The most differences start to arise when we start the process of moving that sentence externally. We apply reductions and linearization which are language specific.
The intersection between HPCT (Hierarchical Perceptual Control Theory) and Operator Grammar explains the details pretty well. And Zellig Harris presents a complete analysis of English Grammar in "A Grammar of English on Mathematical Principles". It is a very different way of looking at language grammar, which makes much more sense to me biologically he spends significant time showing it can also reproduce traditional grammar. I.e. Traditional grammar could be viewed as a change of basis from Operator grammar.
Take a look at An embodied grammar of words[1] for an overview.
[1] https://www.researchgate.net/publication/324362482_An_embodi...