To answer the OP’s geometry question, its easy to see if you consider that for any given place on the globe and point in the earths orbit, if you move the earth to the opposite side of the orbit, those points that were in night become day and vice versa. So asside from all of the caveats expressed above, over the course of the year each instant of time in day has a corresponding instant in night 6 months away.
I agree with the speed of the Earth re: perihelion and apihelion will have less daylight at the South Pole than North Pole.
So testing that out, given the OP that sunrise/sunset at the pole occurs at the equinoxes and in 2010 the Vernal Equinox is 20 March at 17:12 UT and Autumnal Equinox is 23 September at 3:09 UT, I get approximately 186.5 days of sunlight at the North Pole with 178.5 days of dark. Reverse it for the South Pole.
I figure I could put all of the assumptions from the previous poster into a single post.
If we assume:
a. a year is 365.000… days long
b. the Sun is a point source
c. the Earth is a sphere
d. Earth’s orbit is circular
and
e. sunlight is defined as a binary event (no counting lumens),
then yes every spot gets the same amount of daylight.
So in conclusion, no every spot does not get the same amount of sun.