When the team are examining the dead man on the rooftop, his facial expression changes with almost each shot, ranging from a smile to a near-pained expression.
Stella sees a piano in a man's apartment and plays a C scale on it, and when she gets to the top C it doesn't play but makes a banging sound because the strings are missing for that note. This makes her realize that the piano was owned by the killer who used a piano string to kill someone, but the lab said it was an A string, not a C string, so what the lab said and the note Stella played don't match. Plus, the C that Stella played has three strings, not one, so there should have been two more strings, so when Stella played the note it wouldn't have sounded very different than normal and wouldn't have just made a banging noise.
The Statue of Liberty is actually a National Monument, administered by the National Park Sevice, and thus under federal authority, placing it outside NYPD jurisdiction.