Hi fmf rc,

Let's answer your questions in order.

1) AMA is the national organization that oversees model aircraft use. You can become a member of AMA by paying yearly dues. They publish a periodical, provide flight insurance, and oversee the operations of local flying clubs. There are AMA sanctioned fields troughout the country where people can fly their planes and/or helicopters. If you want to fly at one of these fields, you must be an AMA member. As far as I know, there are two AMA fields in NYC. Floyd Benett Field in Brooklyn is for R/C aircraft, and Flushing Meadow is for control line aircraft. AMA insurance covers you in the event that your plane hurts someone or damages property. You don't have to be flying at an AMA field to be covered, but you must be in conformance with all local laws.
2) The best way to find out whether or not you can fly your plane at the field in Flushing Meadow is to contact the Parks Department. (Good luck in trying to find a person who will understand your question, let alone give you a correct answer.)
3) Technically, NYC requires a permit for any activity in any of the public parks. If you want to set up a volleyball net and play a game of volleyball, you technically need a permit. If you want to play soccer, you need a permit. If you want to fly your plane, you need a permit. However, use of any R/C vehicle, whether car, boat, plane or heli, is not permitted in any of the parks except in the designated areas as mentioned in Item 1 above, so you cannot get a permit for Kissena or Flushing Meadow for your R/C plane. If you fly your plane without a permit you are violating the law and you can be fined. You are also not covered by the AMA insurance.
4) The Parks Department Police will issue you a warning for your first infraction. If you disregard their warning, they will give you a summons for the second infraction. (I believe its around $100). The fines become significantly greater with each infraction.

For the most part, if you fly at a time and in a way that does not put others at risk, they will leave you alone.