Lol, thanks guys. As for application software, I work for a large, well-known private university and our system was designed...no joke...in the 1980s. It's entirely text-based, MS DOS style, where you have to tab around to different fields. It's embarrassing, frankly. Ours is probably one of the worst, but that might give you an idea of what you're dealing with.
NYU GSAS sent me an email a couple weeks after I applied with an ID number and a link to their materials tracking system (http://www.nyu.edu/gsas/request/smts.html). That said, it's incredibly unhelpful. It will tell you if they have received GRE scores or anything else that wasn't uploaded with your application, but there's nothing that tells you if your app is complete or where it is in the whole process. Good luck!
I work in financial aid. As a grad student you are always independent on the FAFSA. If you're an undergrad, there are a number of things that will make you independent (age 24+, veteran, married, etc), but if you are a grad student you are automatically independent.
As a grad student in the upcoming year, you're automatically considered independent. The form will only ask for your information, and you should only report your income, assets, etc. Whether or not your parents claim you as a dependent doesn't matter. As long as you're a grad student, the Dept of Ed sees you as independent.