I applied online, and sent a copy of my cover letter and resume to HR to try to stand out a bit (who knows if that worked, or it was just thrown out...) I got a call from HR that I would be receiving an email to take the career battery, then I got another call to set up a phone interview which was very short. This was only general questions (Why NYP, why nursing). I got a call at the end of March saying that they wanted me to come for an interview with HR in April. It was mostly behavioral questions like tell me about a time when you had to change the way you were going about something with a patient. or tell me about a time you had a very difficult patient, and what you did. They also asked some general questions like why NYP, why (the specific unit)... After my interview there I met with the unit manager, and a nurse educator. I think it went well, I got a call back saying they want to extend an offer, but I am hesitant to be excited without a offer in writing. (see post: I want to curl up in a ball and cry)
To answer your other question, I do have strong recommendations. This usually doesn't matter until the interview, and after though. I also have had two jobs as a tech, which I think helped me also.
Let me know if you have any other questions.