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You don't have to take the subway; it's 2 blocks. Penn Station is at 34th Street between 8th and 7th Avenues, and 520 8th Avenue is between 36th and 37th Streets.
Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to 53rd Street-5th Avenue. Fifth Avenue is one block west of Madison Avenue.
Park Avenue is about half a mile east of Penn Station.
Take the Downtown 1 train (of the 1-2-3, the red line), 1 stop, from 34th Street-Penn Station to 28th Street (at 7th Avenue).
You don't need to take the subway to get from Penn Station to 28th Street and 8th Avenue (which is in the Chelsea neighborhood, by the way, not in Greenwich Village or Greenwich Connecticut). Penn Station is at 34th Street between 8th and 7th Avenues. So you can just walk there.
Walk west to Lexington Avenue and north to 53rd Street, and take the Downtown E train (of the A-C-E, the blue line) from Lexington Avenue-53rd Street to 34th Street-Penn Station.
Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to 5th Avenue-53rd Street. Then walk 2 blocks south on 5th Avenue to St. Patrick's (on 5th Avenue between 51st and 50th Streets).
No, the AirTrain takes you to and from the subway, and the subway takes you to and from Penn Station. Take the AirTrain to the Howard Beach Station, and get the Manhattan-bound A train (of the A-C-E, the blue line) to 34th Street-Penn Station.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to Canal Street (at 6th Avenue, aka Avenue of the Americas). Then walk 1.5 to 2.5 blocks down 6th Avenue (depending on which subway exit you use) to 6th Avenue between Lispenard and Walker Streets.
The American Girl store is at the corner of 5th Avenue and 49th Street. Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to 5th Avenue-53rd Street. Then walk 4 blocks down 5th Avenue to 49th Street.
W. 38th St. and 12th Avenue to Penn Station is about 1 mile.
No. You would have to transfer at another station that has access to it.