They have a ton of rights as a New York tenants. Here's the Web site http://www.tenant.net/ of a well-known and very scrappy tenants' rights group. They are in fine shape unless the new lanlord want them out, and if he does, he need to jump through quite a few legal loopholes to do so.
In my opinion nothing changes. The new owner must honor existing lease agreements for the term specified and at the rent specified.
When the lease expires, normal renewal practices are in effect (i.e., if the building is rent-controlled, the same requirements apply as if they were renewing under the old owner).

Nothing will change except the address they mail the rent to.

Yes, the Sponsor may sell any apartments he owns -- occupied or not -- to anyone, without Board approval. The tenants will almost certainly stay in the apartment until they die, since their rent is typically hundreds of dollars a month less than operating costs, much less market rent.

Where this is going to get sticky is the next round, when the investor who buys the rent-controlled apartment eventually sells it to someone else. Does *that* sale require Board approval? It depends on the definition of "Holder of Unsold Shares," which is still in some dispute. Check out the decision in Kralik v. 239 E 79 from 2005 and the alternative view in Sassi-Lehner v. Charlton from 2007.