It's actually quite well known that the birth of hip hop took place in an NYU dorm room right near Washington Square Park in Greenwich Village.
Off the top of my head, the only interesting cultural thing to do up in Harlem is to check out the drum circle in Marcus Garvey park.
http://www.nytimes.com/2008/07/06/ny...pagewanted=all
Read this book if you want to learn more about both jazz movements and other interesting pop culture tidbits.
http://www.amazon.com/Hip-The-Histor.../dp/0060528176
I don't which neighborhoods are dangerous.