I hope I don't bother you.
Of course you don't. Answering questions to the best of our knowledge is our mission here (actually, nobody forces us to do so), so we are always happy to be of help.
Jack is both a good student and a talented soccer player.
Just like "not only/but also" in your example (6) further above, "both/and" are correlative conjunctions, and this makes the whole structure a single (though compound) NP. "both" will be a determiner only when modifying a plural noun and meaning "the two of them": Both students are good.
7. Spend a day in the park, with flowers and green spaces.
"with flowers and green spaces" is a non-restrictive postmodifier for the noun "park" ("in the park" is a prepositional phrase, formed by a preposition "in" and an object, "the park," modified by the "with..." phrase). I say the modifier is non-restrictive because the information provided does not define "park" but is only additional, as if all parks had flowers and green spaces. That phrase is not an adverbial because it does not stand alone, since this does not make sense: Spend a day with flowers and green spaces. It would be a separate adverbial if we said, for example: Spend a day in the park(,) with friends and family.
8. All our products are made of the best material, completed with the great work of our staff.
I don't think this is a good sentence. "completed" is not adjectival but a passive participle (it refers to the action of being completed), and, that being the case, I find that the preposition "by" (meaning "by means of," not "through the agency of") would be required:
8'. All our products are made of the best material, (and are) completed by the great work of our staff. (I think adding "and are" would be advisable to avoid ambiguity.)
9. A popular spot for the youth, the community center is a great place to hang out, organizing a wide range of activities throughout the year.
"organizing a wide range of activities throughout the year" is a reduced relative clause ("which organizes..."). Because of its position, it seems to refer to "place." However, it could be easily moved to the front (which I find better than the original position), in which case, just like "a popular spot for the youth," it could be said to refer to "community center" although it could also refer to "spot":
9'. A popular spot for the youth (and) organizing a wide range of activities throughout the year, the community center is a great place to hang out.
10. This house has bigger windows than any other in the street.
The NP is of course "bigger windows than any other in the street." The "than" phrase complementizes the comparative adjective "bigger." Notice that you can say:
10'. The windows of this house are bigger than those of any other in the street.