Hi, Here is a link to a blog post I wrote a while ago about the process.
Coops are strict with financial and typically require a minimum down payment of 10%, very rarely 5%. Plus, they want to see you have at least 6 months to a year of reserves in the bank AFTER paying 3% closing costs. It doesn't sound like you will qualify for a coop.
Since you are a first time homebuyer, look into a SONYMA loan which allows for a 3% down payment and you can finance your closing costs. if you need a mortgage broker referral please feel free to reach out.