Hmm. Did I have to land? If not:

North: a few miles south of the North Pole on a very atypical JFK-NRT route that went almost directly over the pole, down into northern Siberia then turned back East to get to NRT.

If if I had to land probably Helsinki

South: Hobart, Tasmania - not Cape of Good Hope, which is probably second.

East and West: Not sure how to count it. Been on a few RTW trips. One leaving NYC going east and several leaving NYC going west. The farthest west I have gone from NYC is DC if I count a US city before NYC or Dublin if I count a non-US City. Farthest east from NYC going east before continuing to NYC is Hong Kong.

In trips where I left NYC, went somewhere and turned around back to NYC. Farthest west from NYC was Singapore with Kota Kinabalu, Borneo as a close second. Going east from NYC and returning the farthest was Mumbai.