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 2 or 3 trains (of the 1-2-3, the red line) from 34th Street-Penn Station to Wall Street (at William Street).
Yes, the A express train goes from Penn Station to Chambers Street. It's 4 stops (unless you are traveling at night, when the A runs local). The Chambers Street station on the A and C trains is at Church Street.
Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to Court Square.
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.
Take the Uptown 1-2-3 (the red line) from 34th Street-Penn Station to 42nd Street-Times Square, and transfer there to the N.
Penn Station is at 34th Street so it is 20 blocks away from 54th St. You would take the subway two stops north.
They are both on the New York Subway system, but you'll have to transfer. The best way to get from Grand Central Station to Penn Station is to take the "S" train to Times Square and then take the "1", "2", or "3" trains heading Downtown. The 34th Street stop is by Penn Station. The subway map calls the A, C, E 34th Street stop the Penn Station stop, but the 1, 2, 3 34th Street stop is just as close; walking from the "S" train to the A, C, or E would take awhile since there is a lengthy walk from the S stop on 7th Avenue to the A, C, E stop on 8th Avenue. It's not worth that.
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).
The A train (of the A-C-E, the blue line). Take the Downtown A train from 175th Street to 34th Street-Penn Station. Entrance to the 175th Street station is included inside the bus terminal complex.
Probably the best way to get there is to take the C or E train from Penn Station to the 23rd street stop or the 1 train to the 28th street stop. However, to be completely honest, you could walk from Penn Station to 208 W 23rd Street in around 10-15 minutes.
Take the Uptown 1 train (of the 1-2-3, the red line) from 34th Street-Penn Station to 66th Street-Lincoln Center (at Broadway).