DO NOT BRING THE CAR At best, you'll be having to move it from side to side of the street twice a week... and that's only in the outer boroughs or "less desirable" parts of Manhattan - more likely, you'll pay about $400 + 19% tax for parking. Unless you are planning to do a lot of dancing NJ, it's just not worth it... and if you are planning to do a lot out in NJ, then stay there instead.
In terms of where in the city, much dance activity used to be concentrated in the 20's and 30's - within walking distance of the Empire state building. But with Ballroom on Fifth not currently dominant (the new one is just opening up on 37th street), there's been more activity up at Dance NY in the west 50's, and at Studio 101 just south of Canal in Chinatown, as well as at other places in midtown such as Stepping Out on 26th street. Still, these are all within an easy subway ride of each other. Hardly any of the leading teachers are studio specific at this point, instead moving around from day to day to avoid group classes and the like, so they will tell you where the lesson will be when you schedule it - though there's always a chance you might get a later call changing the location.
In terms of where to stay... well, where you find something suitable and affordable. You'll want to spend a lot of time on Craigslist looking at short term rentals from people going on vacation or something. Location wise, in Manhattan Hell's kitchen (west 40's), Chelsea (west 20's), and (if you are lucky) Murray Hill (middle to east 30's) are all fairly convenient to dancing. I'd also look at parts of Queens with good subway service such as Astoria and comparable parts of Brooklyn, though I don't know what they would be. Unless you are routinely dancing until 1 in the morning, being a little further out is not a big deal - and if you are in the city late on weekends, the subway still runs, it's just a longer wait for a train to queens or brooklyn (some of the public transit to NJ does stop though).
