Alright, first things first...
Arcades: Your selection is quite sparse. Perhaps the best is one in Chinatown, which you'll find on Mott Street. They have a good selection of fighters, a DDR set up, and some old classics.
Plus there's Broadway Arcade on 42nd Street and 8th Ave. It's one of those glitzy entertainment centers that uses cards instead of quarters. It's also pretty expensive, so I might suggest you simply go to the bowling alley located at the Port Authority which is very close by, which has a few machines in the lobby.
I've heard of a new place called Neutral Ground at 122 West 26th Street. There's a tourney going on soon. You'll find more info here:
http://ga-forum.com/showthread.php?t=4966
Speaking of...
Game Related Events/Hang-Out Spots: Once again, not much to speak of (New York is pretty lame when it comes to video game related stuff to see and do). But there is Gamer's Nite Groove. There's one every few months or so. For more info, just go to:
http://game.newyork-tokyo.com/
I've been to a few and they're alright. I honestly don't know why I even give them any press since one of the organizers is a major douchebag.
They used to hold it at a place called Game Time Nation:
http://www.gametimenation.com/
There you can rent games and play them on their big tv's and comfy couches. It's a pretty nice atmosphere actually, though pretty pricey. Which leads to...
Game Stores: You'll find Gamestops and EBs all over the city, but there are a few mom and pop stores.
The best has to be Game Express on West 32nd Street between 6th and 7th Avenue. They have a huge selection of new and old titles and many great deals. Sale people are okay.
The best known Chinatown store is J&L on Elizabeth Street. Go there to window shop but don't buy anything; they have a huge assortment of import titles, and games for older systems, but everything's way too overpriced. Plus, don't trust them to do a mod job cuz they'll fuck up your system.
Go across to street to the Elizabeth Street Mall instead. In the lower level you'll find Penguin Village. They have a much smaller selection, but they're prices are almost always cheaper than J&L.
Also well know is Multimedia at St. Mark's in the East Village. They have an extensive selection of classic stuff, all the way to the 2600. They also have the biggest asshole in the city working there (this is a very well known fact).
Often of not, you just can't go wrong with Best Buy, which is located at West 23rd and 6th Ave. There's another location at 86th, right off the 6 train (can't remember the Avenue).
Hope this helps.