First, if "New York" appears twice on the same page, in the Index window twice.
That's not a problem. The entries in the Index Window point to where the index marker is, and it's not a preview of how your actual index is going to look.
Second problem is that when I select "Generate Index" it gives me a listing for "New York" but no page numbers whatsoever.
Now that is a problem. There were a couple of index-related fixes -- can you check what version of InDesign you have? Look in the menu "About InDesign" for that.
I'm on 7.0.
Well, then you need to update! Latest patch level I know of is 7.0.4.
I don't know if it solves your indexing problems, but you sure can try and see what happens.
Does anyone else know whether this is a software problem solved by the latest patch or the dreaded "operator error"?