Typically, incoming airliners will fly directly down onto the runway on the ILS, after being vectored onto the latter by ATC. Outgoing traffic is usually routed out along well-established routings, often to a nearby VOR beacon, and then onward (on a flight plan given during clearance delivery.)
As regards 'the questions', I assume you are talking about the military Warning area south of LI. These are all over the waters off the East Coast, and are used primarily for mil flight operations. Maybe you are suggesting that the Navy was conducting surface-to-air missile firing operations at the edge of W-105. I guess anything is possible, but as I said at the start of this discussion, I think it's higly unlikely that the Navy would be irresponsible enough to deliberately fire heat-seeking missiles up into the air that close to JFK operations (and that close to the Hampton VOR and airway V-46). An accidental firing? Maybe.
One of the problems with testing at White Sands and places like that is that they don't have the same "noise" problems as would be encountered near populated areas. These missiles have to be able to operate in high noise areas because that's where they would most likely be used. That's why they test in places like W-105.
ML/NJ