I am hoping that you are not thinking that being a gay couple will make a difference here. What I find is important when buying co-ops in Manhattan, if that is the route you choose, is having an agent who is familiar with the building so they are aware of how the board makes their decisions. If they are stricter, then the seller might be less willing to take your offer. From my experience, this has been the case, even when I have been working with the couple, and had a commitment for them before contract were even signed. This was 7 or 8 years ago, and things have changed, more in terms of whether it is a buyers or seller's market, than anything else. If it is a seller's market, they tend to be a little more picky.
The important things are whether you are qualified, how well, meaning more like the 30% debt-to-income ratio in some buildings, as apposed to the 38%-45% that you can get away with from a lender. Also, they may have a requirement as far as post-closing reserves. Depending on your loan size, and what bank your loan is sent to, there are differences there as well. In that price range, you are unlikely to find a co-op building that allows 10% down. Even 20% is not what is allowed in all buildings. Not knowing what you are thinking about, you might not have enough to look at some buildings if they require 25% or 30% down, if that leaves you short of reserves.
I would be happy to have you contact me to review all of the details, and see what is best for you. My company is a mortgage banking company that also gives us the flexibility of brokering loans to various banks if their product and rate is better than what we can offer. Although my office is on Long Island, I am very familiar with the NYC condo and co-op market. I have been in the business for over 13 years, and have seen its ups and downs, both in terms of rates, programs, availability of money, you name it.