what functions can "all the way" perform in correct grammar?

The phrase "all the way" appears to be a modified noun. "the" is a definite article (determiner) introducing a noun. "all" is a predeterminer informing us of the quantity of the following noun. However, this noun phrase most often functions as an adverb.

As in: He ran all the way... They fought all the way... and I have not been able to include this expression in a sentence diagram. :?: I read in one source that the expression is an idiom - which is defined there as a group of words which cannot be understood by applying standard grammar rules. This phrase, by itself, seems to be quite understandable, but its usual function seems to be ungrammatical. Can someone clear up my confusion?

We are talking about slang so standard usage and grammar do not matter much. Since your native language is American English as your profile states, your opinion is at least as good as mine, if not better. So please tell us what you think about your question and maybe we can get an interesting discussion going.

*Not a teacher* Responding to probus. I believe you must be correct that it is slang or an idiom because I can find no way to fit it into a standard English sentence using the rules of grammar that I know - or think I know. Yet it is often used in ordinary conversation and writing and I've always understood the meaning and assumed it was grammatically correct - until I recently tried to see if it actually does. I've never before seen any discussion of it.

In its entry for way thefreedictionary.com says all the way is an adverb and an idiom meaning from beginning to end or completely.

That seems to account for its grammatical function in all the examples you gave, and also in going all the way, which was widely used when I was a teenager, but seems to have died away somewhat.

Even when used as an exhortation or encouragement, as when someone shouts out all the way baby it still seems like an adverb to me. The person addressed is being encouraged to do whatever he is doing to as great an extent as possible.