Most hotels will allow this, but wont let you know if they will grant your request until the morning you are supposed to check out. I stayed at the Hotel 30 30 the last time I was in NY and we were allowed to check out 2 hours late for no charge, but it totally depends on the folks coming into the hotel that day. If they need your room - no late check outs will be granted.
Some hotels will also give you an extra half day for a fee, but again it all depends on their circumstances.
As far as what to do with your bags, you do not need a late check out to leave your bags at your hotel. In all of my experiences, they will hold them for you even after you have checked out. You may also be able to leave them at the pier and then run around the city for a bit.
Hope that helps