Here's a starting point : <$2000 downtown Manhattan. You'll get much more if you're willing to extend to Brooklyn/Queens, or spend a little more (or go for a share).
http://streeteasy.com/nyc/rentals/downtown-manhattan/rental_...
http://streeteasy.com/nyc/rentals/downtown-manhattan/rental_...
Most of the questions you asked have been answered already. So I ll give you some pointers about the housing question.
1. Craigslist. - http://newyork.craigslist.org/ 2. Padmapper - http://www.padmapper.com - maps all the apartments on craigslist. 3. http://www.nybits.com - for apartment listings from building managers.
The average "decent" studio apartment in Manhattan will cost you anything from $1000 - $3000. It might be a little lesser if you get a bigger place and share. Living costs are very high, but the fact that you could go shopping for groceries across the street at 3am makes up for it.