Scooters are cool. But you might as well get a motorcycle.
I recommend this one.
It's Italian, just like your Vespa, so you'll very much be riding in style.
To answer some of your other questions.
1) I don't see how this makes a difference. Riding a scooter or motorcycle is inherently risky... regardless of previous injury. I don't see how this injury makes you any more susceptible to further injury.
2) Just bundle up and you'll be fine. It kind of sucks riding in the rain, though.
3) Good gloves and a thick jacket are important. At low city speeds you don't need to worry about full leathers or moulded body armor or anything like that. Just tough thick jeans and a thick jacket that won't flap around too much.
4) Can't help you there. Just don't know.
5) Get the Ducati. Take it to the dealer a lot.
6) Take a safety course. MSF?
7) Get it now; take the course, learn how to ride before you enter NYC traffic.
Finally, you will enjoy the scooter, even though I've been harping on the Ducati. Just learn how to ride safely.