New York doesn't possess anything in this sentence. It is being used as an adjective to qualify "office." That's all.
To be frank, I'm not really sure what you're asking with your other questions. You can use "New York office" whenever you want to describe an office that's in New York. A valid usage of "New York's" might be something like:
It's a very tall building, but not as tall as New York's Empire State.
The Empire State building is a cultural icon strongly associated with New York, so although New York as a city (or a state, depending on context) cannot "own" things, the building "belongs" to New York. It is a bit of a metaphor in this usage.
Other things that you can apply possession of a place like New York to would be: its laws, history, officials, etc.