CPLR 5519 is tricky. Did the judge who issued the judgment and order of eviction set an amount to be paid (5519 [a])? Did it specify an amount for an undertaking?
The most important section for you is 5519[a]. That provides for a stay if you pay an undertaking in an amount set by the court, and, if you lose on appeal, you must pay reasonable use and occupancy, from the date of the notice of appeal (March 9) to the date you give up possession of the property. You will also be responsible for interest for the same period, which in NY is 9%.
In your order to show cause, did you ask the judge to fix the amount of the undertaking? It's usually twice the amount in dispute.
I'd suggest you have a lawyer handle this for you from here on out. You're in murky territory, one that some lawyers find hard to navigate. And please don't write your own brief. I've never seen a good one written by a layperson (and I've seen terrible ones written by lawyers, too).
DISCLAIMER -- This is not legal advice. It is for general information purposes only. No attorney-client relationship is hereby created.