If you don’t have your status check password a week or so after the deadline, I’d definitely call NYU to check. If they emailed it, then they’ll just re-email it. If they didn’t email it, then they will be able to check and see what’s going on.
As far as missing info, here’s my story. I applied on May 1 and received my status check password a week or so later. At that time, my status check showed the generic message. Then, during the first week of June, I was notified by email that NYU needed a transcript. When I did a status check after that, the site indicated that my app was incomplete and it also specified exactly what NYU needed from me. It only took me a few days to get the papers to NYU, and it took NYU about 2 days to register it. After that, my status check site showed that all the info they asked for was received and it also said the original generic message. My admit notice came a little under 2 weeks later.
Basically, if NYU wants you, you’re not going to be kept out for trivial things like slightly late letters of rec. or a missing doc (if you can get it to them quickly). Don’t sweat this stuff.