1.
They may have the right to be armed (as well as you and me – 2nd amendment) but they shouldn’t carry weapons. Besides that in New York it isn’t that easy to obtain arms, I believe that as an armed patrol force it may be illegal. But, it isn’t a matter of legality, it is a matter of safety. You must go through extensive training to be a cop. Just like you have a weapon, the criminal does and he’ll shoot.
2,
Yes. It should be left to the Police dept. Lives should not be placed in danger because of some money. In Halacha, you are not allowed to call police on Shabbos when all it involves is money (only at a Chashash Pikuach Nefesh). They have no business being at every other investigation or car accidents. There’s nothing they can do and should stick to basic patrolling and leaving the rest up to the cops.
3,
It is 100% Assur!
Even in a place of Pikuach Nefesh (a time where everything is permitted except 3 Aveiros) it is Assur. You are not in control of your Guf and you cannot decide to play games.
4,
You cannot deny that it was an heroic move (if I jump down the Empire State Building it is heroic as well). However, it was dead wrong!