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There is one - the north pole. Note that you could go 1 mile south, 1 mile west, and one mile back north to end up on the north pole where you began your trip.

Similarly, you could do the opposite directions from the south pole. Go 1 mile north, then 1 mile east or west, and go 1 mile back south to end up on the south pole again.

(Note that moving E or W from a point near the poles involves a curved path rather than a straight line.)

No, there are an infinite number. At any point on the circle that is 1+1/(2 x pi) miles from the south pole, you can go 1 mile south, 1 mile west will do a complete circuit of the South pole and get you back to where you were after the South leg, and then 1 mile North will get you back to where you started.

And at any point on the circle that is 1 + 1/(4 x pi) miles from the South Pole, you can go 1 mile South, 1 mile west will do 2 laps around the pole, and 1 mile North will take you back to where you started ... etc.

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Q: How many points are on the globe that you can go 1 mile south 1 mile east and 1 mile north and still end up in the same point you started at?
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