Take the Downtown Q express train (of the N-R-Q, the yellow line), 2 stops, from 42nd Street-Times Square to 14th Street-Union Square. You can also take the N or R but they are local trains so it will take a lot longer.
Transfer at 14th Street-Union Square to the Downtown 4 or 5 express trains (of the 4-5-6, the dark green line), 1 stop, to Brooklyn Bridge-City Hall. Again, you can also take the 6 but it is a local train and will take much longer.
You'll exit the subway at the intersection of Chambers and Centre Streets (at the northeast corner of City Hall Park). Walk south on Centre Street (along the east side of City Hall Park) until you reach the entrance to the Brooklyn Bridge Promenade (about 1/3 of the way down the length of the park).
Take the Downtown N-R-Q (the yellow line) from 42nd Street-Times Square (at 7th Avenue) to 14th Street-Union Square.Transfer at 14th Street-Union Square to the Downtown 4 or 5 express trains (of the 4-5-6, the dark green line), 1 stop, to Brooklyn Bridge-City Hall.
Penn Station is a train station. So, obviously, there are plenty of trains that stop there. If you were referring to New York City subway trains, then, the 1-2-3 (the red line) and the A-C-E (the blue line) stop at Penn Station.
The old BRT line ran over the Brooklyn Bridge and just barely into Manhattan, to its last stop, but that was around a hundred years ago. I don't believe the BRT station was actually on the Brooklyn Bridge, but rather just beyond it, in Manhattan.
Take the shuttle (called the AirTrain) from inside the JFK terminals to the Howard Beach Station of the A train. The A train goes through Brooklyn. JFK is in Queens, and Brooklyn and Queens are two halves of one island. Queens is the north side, and Brooklyn is the south side.
Much of your question depends on whether you're driving or walking, where you start on the Brooklyn Bridge and where you end up in Central Park. If I walked from the Manhattan side of the foot of the Brooklyn Bridge to the corner of 5th Avenue and 59th Street, I'd walk about six miles.
No. The subway trains to and from Brooklyn run on their own elevated lines. They do not use the Brooklyn Bridge. The bridge is for cars, buses, trucks, and pedestrian traffic only. There used to be trolleys that ran across the Brooklyn Bridge, but that was only from 1898 to 1950. The old BRT line also ran over the bridge and just barely into Manhattan, to its last stop, but that was around a hundred years ago.
You can take the Downtown R train from 34th Street-Herald Square (at 34th Street and Broadway) to west Brooklyn (the neighborhoods of Downtown Brooklyn, Boerum Hill, Gowanus, Park Slope, Greenwood, Sunset Park, Bay Ridge and Fort Hamilton).
You can get an underground train and an overground train from London Bridge.
Unfortunately the number 2 train does not go to Rector St in Brooklyn. There is not a train that goes to Rector St. However there are many buses and taxis that do go to Brooklyn.
Gasconade Bridge train disaster happened in 1855.
No, the N train (of the N-R-Q-W, the yellow line) does not stop at Penn Station. But it stops at Times Square. Take the Uptown 1 train (of the 1-2-3, the red line) one stop, from 34th Street-Penn Station to 42nd Street-Times Square, then transfer at Times Square to the N.
Take the Uptown 1-2-3 (the red line), 1 stop, from 34th Street-Penn Station to 42nd Street-Times Square. Transfer at Times Square to the R.