Yes you can photograph from the Brooklyn Bridge. So long as you are not doing commercial photography, there is no problem. I know this for a fact since my girlfriend was there the other morning photographing Peregrine Falcons that were about to fledge from the nest (Brooklyn side arch). She had her 500mm lens/tripod, etc., and had no problems.
Sunrise from the BB is not very good...the morning sun on lower Manahttan is nice though. My favorite location for sunrise is Bow Bridge in Central Park...but again, the sun is at your back for the best shots. If you want to catch the actual sunrise itself you will need to get in a tall building in Manhattan and shoot toward the east. If you want the nice, warm lighting of sunrise, then anywhere is good...
Robert DeCandido, PhD (a displaced NYker for a few months)
Post-Doctoral Fellow in Ornithology
International Bird Research Centre in Eilat, Israel