Once you send in your $1000 deposit, NYU will send you the ASCOE form to fill out. Once you return the form, they will send you the I-20 in 2-3 weeks. The info on this form is reported on your I-20 and you will be required to present financials. Once you receive your I-20, you will need to present this and some other applications to the embassy.
Since I am a Canadian resident, I know everything that is required for here, but I am not sure if it is the same or different for US residents. You do not have to go back to your home country to apply. The most important aspect of your application is financial support/loans for your education.
They do not have on-campus housing for married couples. You can go through their website for on-campus housing if you are single, or if you will be living separately from your spouse. I would recommend off-campus housing. There are many apartments nearby.