I went to a Yankees game, saw Blackbird with Jeff Daniels and Michelle Williams (forget the theater), Smalls Jazz Club in the West Village, Nom Wah Tea Palor in the Chinese district, Diwanekhaas in the financial district, walked the Brooklyn Bridge then around Brooklyn. I mention the different locations and where I went because I walked everywhere to and from then around if something caught my eye. The only time I didn't walk is when I took the subway from Smalls Jazz club was back to the apartment I was staying at which was near W 48th and 8th.
As I stated that was my first time visiting, so I am not really sure how that covers NYC from a local perspective - I was certainly tired after 5 days. However, I did see police everywhere I went, except at 2am when I was walking from the subway back to the apartment.
Every place you mentioned I would expect significant police presence either because of the venue or locale. Since the Germans 'artists' swapped out the flags atop the Brooklyn Bridge there has been a permanent and rather disrupting police present on the bridge itself.
As I stated that was my first time visiting, so I am not really sure how that covers NYC from a local perspective - I was certainly tired after 5 days. However, I did see police everywhere I went, except at 2am when I was walking from the subway back to the apartment.