Hi first time home buyer.
To answer your question gently. Without knowing all the facts, I would have to say that you may not qualify for this building or other coop buildings. Most coops as you already know do require 20% down payment, and I think they are even stricter in Today's Market. A main reason why New York is still strong is due to Coop buildings and the financial requirements they put on a buyer. This is not the time when they will be more relaxed, quite the opposite.
If you find a coop that only requires 10% down payment, you will face into another opponent and that's being the BANK. Most banks now require 20 to 25% down payment. with a 10% down payment your interest rate would also be on the higher side and you would want to put more money down.
Would it be possible for someone to gift you money? Please keep in mind that the coop board will want to see liquid cash in your bank account after the purchase and closing costs.
The building which you listed has 4 apartments, and they have a very low deductible of only 38%.
Can you wait, rent, save and purchase at a later time?