9''. A popular spot for the youth, organizing a wide range of activities throughout the year, the community center is a great place to hang out.
Although its function is adverbial, the NP "A popular spot for the youth" is a sort of loose subject complement within the predicate (notice it comes from "Being a popular spot for the youth") which does not have sufficient syntactic hierarchy to accept the addition of a non-restrictive participial post-modifier. Even at first sight that comma before "organizing" looks really confusing. I think we should eliminate it. Now that I reread (9'), I would add "and" only if "being" were used at the beginning so as to ensure grammatical parallelism:
9'''. Being 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: The house is only a 5 minute bike ride from the school.
--> I would say "only a 5 minute bike ride from the school" is one NP? Is that right?
Yes, it is one NP with "ride" as its head.
12: It is the dream of my father to travel to China.
Only "of my father" is a postmodifier of "dream." "to travel to China" is an infinitival clause functioning as the real subject anticipated by "it." Notice we can say:
12'. To travel to China is the dream of my father.