"There are seven books on the bookshelf."
Placing "nana satsu" before "no" makes the 7(books counter) the possessor of the books. So its like saying the (in my attempt at translating this), In bookshelves the 7's book there are. Like the 7's have books.... not the other way round.... in fact thinking about it.... probably wouldnt work the other way anyway either, since that would say the book possesses 7.... the books 7, which i would imagine would work if u were talking about pages or something like that
Correct me if im wrong... pretty curious about this now lol
Putting 七冊の本 (nana satsu no hon) is a valid way to say "7 books". The の (no) in this case isn't showing ownership or posession, it's just modifying the noun 本 (hon).
Here is a link that expands further: https://en.m.wikibooks.org/wiki/Japanese/Grammar/Counters
It's not incorrect, but it's also less frequent to phrase it like this if you are just trying to say new information, so most of the time it could be interpreted as a delimitation, kinda like saying "the seven books are in the bookshelf" in comparison to just saying "There are seven books on the bookshelf" in English. Not necessarily exactly like that, but that's the feeling I get from reading both ways.
If I understand correctly, the さつ (satsu) part is needed because the objects that are counted in this sentence are books. It has the same role as つ (tsu) when inanimate things are usually counted. Here is an example for comparison:
いす が 七つ あります。>> There are seven chairs. (いす = chair)
本 が 七さつ あります。>> There are seven books. (本 = book)
Notice how the 七つ (nanatsu) part changes to 七さつ (nanasatsu).
As for the あり (ari) part, that belongs to the あります (arimasu) part of the sentence, which basically means "there are".
「が」 Indicates the subject in this scenario. It has other uses, but for now, let's focus on its use as a subject marker. As you may have noticed, がand は are used after the subject of the sentence or as a particle between words. This is why が is not needed where it has been placed since there is no particle needed nor any subject before が。
さつ is a counter for books (depending on which kanji it is, it's a counter for paper money) and you have to state what item you are counting on the bookshelf. The way you structured your sentence seems almost like you're counting the bookshelf and not the books, because the way it works is that you put the item in subject before the particle.
You're not really saying "本" twice in a sense of using the meaning of the character twice.
本だな (hondana) means "bookshelf"
本 (hon) means "book"
I hope this helps.