Why does the earliest sunset time not coincide with the shortest day of the year?
If you follow the sunset timing for the equator you will notice that although the day remains pretty much the same all year round, the sunset moves up and down twice a year. The biggest difference is about a half hour.

This is due to the changing speed of the orbit of Earth. Earth's orbits is slightly oblong. Therefore on the longer parts the planet picks up speed. Which means that although Earth made a full rotation, since it is further in its orbit it is still seeing the sun. 

This happens twice, on both longer stretches of the orbit. However the effect on one is more pronounced than the other. This is because in one leg Earth is speeding away from the sun, which slows it a bit, and in the other it is coming toward the sun.

The longest and shortest days are on both points of the oblong circle but the later sunset is before that point.