Don't do anything crazy until you sign a contract saying you're hired. I would try to negotiate starting remote with intent to relocate within 60 days of start date. From there, you can try to get your relocation fee on your first paycheck and move quickly.

Alternatively, look up everyone and anyone that you know/have known and see if they're in or near NYC. Ask to crash with them until you find a place and say you can pay them for the inconvenience (use salary as confirmation/find alternative collateral).

Early termination would generally be taken out of deposit/last month rent. You can also try to negotiate with your landlord if you're on good terms with them.

Again. Do NOT do anything until the job is secured. Recruiters are generally just middlemen - negotiate directly with the employer/manager.