You have understood one thing correctly: a preposition is combined with what follows it to form a prepositional phrase, and they form a single constituent in the sentence. Generally speaking, words in English govern (ie. control or specify) the words that come after them. In linguistics, we say that English is right-branching, meaning that new syntactic elements come after (to the right of, in writing) the elements that govern them.
Note that there are exceptions, such as adjectives, which precede the nouns that govern them. English is not exclusively right-branching, but it is predominantly right-branching.
But what does this have to do with prepositions? Well, just as a preposition governs the noun phrase that comes to its right, the preposition itself is governed by something to its left. And in many cases, that thing is a verb. English is full of idiomatic combinations of verb + preposition, where the verb requires a specific preposition to follow it, and anything else is an error. To take some obvious examples cribbed from other answers:
I converse with you. [Not to/at/of you]
They rely on the bus. [Not with/to/at the bus]
These combinations are highly idiomatic, meaning that the correct choice of preposition cannot be predicted simply by knowing the general meaning of the words involved. So the people who ask about what preposition follows a certain word are asking a reasonable and intelligent question. The choice of preposition very, very often depends on what came before it.
In this case, the phrasal verb is put down as you have correctly said that the particle can go either side of its direct object
to put down many coats
to put many coats down
The to immediately following that phrase is part of to do the job.
In order to completely hide the horrible colour, I expect I would have to put down many coats of paint.
Now, there is an idiom to put down to, as in "I put the high cost of food down to the price of fuel" which means "I believe the reason for the high cost of food is the price of fuel". Your original example doesn't use this, and it's not a separable phrasal verb (as in your Biber reference) because down to can only come after the direct object. You can tell the difference because the idiomatic put down to is always followed by a noun or noun phrase.
Best Answer
This is a tricky question.
As strong evidence that at least some occurrences of "go to sleep" are using the preposition to and the noun sleep, I note that "went right to sleep" is well-attested, which lines up with "went right to school", "went right to the heart of the problem", and so on; whereas we don't say *"went right to talk to her", or *"went right to see what was happening", or the like.
Another piece of evidence is the related idiom "put <someone> to sleep"; put takes a prepositional phrase or the grammatical equivalent ("put <something> in the trash", "put <someone> outside", etc.), not an infinitive (*"put <someone> to be <participle>", *"put <someone> to talk to her").
Conversely, I can't find any evidence that any occurrences are using the to-infinitive to sleep; I tried a few different kinds of searches:
So I would tentatively suggest that "go to sleep" only ever uses the preposition to and the noun sleep, even though in the vast majority of occurrences there's no way to definitively rule out the parse with the to-infinitive to sleep.