Day 1-4: Row NYC
Day 9-10: The Time Hotel
New York is wonderful. It offers so many things to explore that it is difficult to focus on just a few highlights. Climbing the top of the Empire State Building in the evening is one of them. The view over the city at night is literally outstanding. The bull at Wall Street is also worth a visit, especially because by any hour of the day it is surrounded by Chinese who are grabbing him by the balls. Walking Brooklyn Bridge back to Manhattan is also recommended. Not only because it is an exceptionally beautiful bridge but also because it offers you the chance to walk into the big apple.
In case you need a quick break from the big city, Central Park is just the place to go. You can spend an afternoon in the nature crossing countless bridges and if it would still be 2016 watch people hunting Pokémon, but maybe they are hunting something else meanwhile.
Highline park is also such a place to escape. It is a park built on an old highline track containing plants and relaxation areas and there are always some street artists performing.
Great food you can find in many places but Chelsea market and little Italy are definitely worth a visit. At the Stardust Diner you only get average food but served from the most outstanding waiters and waitresses, because they are all outstanding singers and they recite musicals while they are serving your table.