Shortest day

[HGray]

If the shortest day happens because we are at our furthest point from the sun, why isn't it the coldest day, too?

[incitatus]

It doesn't. The shortest day happens because of the earths axial tilt, rather than because of distance from the sun. If the latter were the case, December would be winter in Australia too, which it isn't.
The reason that the coldest days tend to be later than December is that the earth retains heat ... it takes it a while to heat up and cool down. This is why the longest day is in June, but the hottest ones in the northern hemisphere are normally in July/August.