NYU SPS Capstone Competition

Capstone experiences are academic courses that "bring it all together". So by definition this is an incredible opportunity for your resume where you show how you're applying all you've learned in your program to solve a problem, construct a well developed thesis, and to discover and synthesize new knowledge. The capstone experience should be a highlighted accomplishment on any resume and discussed at an interview. 

Yes but you should also consider finding a faculty member who has similar research/academic interests to your own, as that will help with the development of your project. You can review faculty bios here.

It is possible, but that is something each faculty member and program will want to discuss with you on an individual basis.

All capstone experiences are designed in consultation with the faculty, so yes it would need faculty approval, and you would work in tandem with them to construct a successful project plan and deliverables.