Yes, when they use that shorthand, they are referring to streets in Manhattan, and if they’re First through 10th they are avenues. The avenues, plus Broadway and a couple of smaller fringe streets, run north to south. The streets go up to something like 244th, including part of the Bronx. They generally use the street (or avenue) that the address is actually on before giving the cross street. And you’re right, if simply referencing an intersection, the avenue comes first. They’ll slap an “Ave.” and/or a “St.” on if it’s open to confusion.
Thanks for the clarity and generosity of your replies. I’ve now learned a couple of things I long wondered about but had no one to ask.