There is a report of a captured FW 190 arriving at Wright Field, Ohio in August 1943. However, it seems very unlikely that it would have passed through Bethpage first for an evaluation by a Navy contractor, although I'm sure that it would have been a closely held secret if it did.
As best I can determine, the F8F originated with a memo from Roy Grumman to Chief Engineer Bill Schwendler dated 28 July 1943 requesting a predesign of a small fighter built around the most powerful R-2800 engine available and providing some additional guidelines. It was reportedly the result of previous discussions between those two dating back to at least late 1942.
A predesign drawing by Dick Hutto dated 20 August 1943 indicates that the basic size and shape, including a bubble canopy, of the Grumman G-58 were well established by then.
The story that seems more credible (and supported by contemporaneous documentation) is that Grumman's Bob Hall and Bud Gilles went to England in September 1943 to fly a captured Fw 190. I haven't seen Hall's report, but they were undoubtedly impressed by its speed and maneuverability. It seems likely that they would have returned to Bethpage with the intent to match, if not exceed, its performance and handling qualities with the new Grumman fighter.
It therefore seems almost certain that the basic philosophy that shaped the F8F was not the September FW 190 flight evaluation but the result of 1942 combat experience in the Pacific vis-a-vis the Mitsubishi Zero and the need for a small fighter that had better performance than the F4F Wildcat to operate from the small decks of the newly created escort carriers.
The Navy ordered prototypes of the G-58 in November 1943 and designated it the F8F. The first one flew only nine months later, in August 1944. Deliveries of the first production aircraft were made in February 1945.
However, the first air group equipped with F8Fs arrived in the Pacific just days too late to participate in the war. (In fact, it was first Navy carrier-based aircraft initiated after Pearl Harbor to get that far; most were canceled before reaching fleet squadrons.) Within a few years, it was supplanted by jet fighters and relegated to a training role.