Yes, "He put his book down on the table." makes sense and would be used by a native English speaker, though it would also make sense if the sentence was "He put his book on the table.", omitting the word "down". If the person were putting the object on something higher than waist level where one would have to "reach up" one might add the adverb "up", as in the following examples: He put the book up on the shelf. She put the vase up on the mantle. I put the luggage up on the car rack. However it would still make sense even if the word "up" were omitted, the listener/reader would not be confused. In contrast, one might add the adverb "down" if the person were putting the object on something lower than waist level where one would have to "bend down", as in the following examples: He put the book down on the floor. She put the vase down on the ground. I put the luggage down on the baggage carousel. As with "up", these sentences are still both correct and understandable if the adverb "down" is omitted. By the way, this "waist level rule" (if one wants to call it that) may simply be my interpretation of common English speech, I can't cite any specific grammar references for its usage.

I think what you're asking is whether your analysis of the grammar in these sentences is correct and makes sense? Q 1: He put his book down on the table. My interpretation is that "put (something) down" is a very commonly used verb phrase which is modified by the prepositional phrase "on the table." This sentence is correct in grammar and usage, and it's something I might say any time I needed to talk about putting books down on tables. Q 2: He put his book down on the table. Here I think you're talking about the adverb phrase "down on the table" (which itself is made by the adverb "down" modifying the prepositional phrase "on the table") modifying the verb "put"? That doesn't quite work. When directional adverbs (up, down, over, etc.) are used with a prepositional phrase like this, it implies that all of the action takes place within the place mentioned: - My mother grew up milking cows down on the farm. - Up in Alaska, they have a lot of great seafood and bad roads. - I left my bag over on the other side of the room. - She was already at the top of the ladder when she realized she'd left the box of light bulbs down on the table. "Down on the table" is somewhere that you can leave a box of light bulbs - all the action of being left takes place on the table. But you can't put a book "down on the table," because the action of "putting" starts somewhere other than the table. ....Is that what you were asking? And if so, have I answered your questions?