I agree, NY is a huge place (I'm originally from the Bronx, currently living upstate in the Adirondack area)....you didn't mention which boro you are in, therefore it makes it difficult to narrow down your local choices. I would suggest doing a google search for Italian import stores or delis based on your address (or neighborhood). Then, before running out to the store, give them a call to find out if they carry the items you are looking for.
Many supermarkets have "ethnic" sections where they carry Italian, Mexican/Latin and Asian products. Many supermarket deli departments carry things like prosciutto di Parma, from Parma, and prosciutto di San Daniele, and their cheese departments carry many imported Italian cheeses. As for "fake Italian products", I'm not sure I know what you mean. Do you mean American made "Italian" products/brands? Some of them are just as good or better than some of the imported products.
As for packaged goods, read the labels - there will either be an address of the company in Italy that makes the item, or it will say "Imported and distributed by: "so - and - so USA." Sometimes the labels of imported products are printed in Italian, but some companies change the packaging for the US market.
As you know, true Italian cooking (as in all cooking) revolves around using the freshest, and best quality ingredients you can afford (or grow). Most "authentic Italian" dishes are simple dishes, it is the technique used in the prep and execution of a dish/meal that makes it what it is.
