Owning a coop is not owning real estate, actually, it is owning shares in a corporation, ie. the cooperative. There are instances when people who live in coops are covered by the New York Rental law, but not always. I think you should get a copy of the proprietary lease, house rules, etc, AND if you can locate a copy, The Condo and Coop Bible. The latter is a great intro to how coops and condos work, with particular attention to proprietary leases and such.

In my building, for instance, a shareholder can leave his/her apartment to anyone, but that person still has to pass the board (based on income etc) if he/she wants to live there. Usually heirs just want to sell. There are also instances when other names are put on proprietary leases. As far as I know, as long as someone is actually LIVING in the apartment, there's no way to prevent that, though in fact, there is usually a lot of paperwork involved.

Your dog situation does complicate things. Even if the board is predisposed to your situation, which isn't a given, a misbehaving dog on the part of a potential new resident is a potential problem for the board. This is true even if the house rules are silent.

Good luck. Keep us posted.