EB:

The Hudson (or, as you point out, more correctly "North") River does not touch upon Brooklyn at all, as far as I can see.

Does the "official" Erie Canal go down the Hudson, out into Upper New York Bay, and proceed east-southeast to Brooklyn? Or does it depart from the Hudson between the Bronx and Manhattan and take the Harlem and East Rivers (neither of which are really rivers, technically) to that basin in Brooklyn?

Depends upon where the Erie Basin is located, obviusly. You've made me very curious!