Hi Ethan,
I know for a fact that you don't have to be a resident of either the city or the state of NY to receive both their services and the sliding scale discount.
It takes them about a week on average to receive the blood work results (from an off site lab in NJ, billed separately), the only constraint is the schedule of whoever will be treating you. Most of the times you'll wait for a few weeks (or months when they're busy), and if you're lucky you will manage to take over someone else's canceled appointment.
Please PM me if you have any questions, I'm always glad to assist if I can 🙂