How does egg hatching distance relate to the hatched Pokémon?

Solution 1:

All first evolution stage pokemon except for legendaries can appear. This is spread throughout the eggs as follows:

Chart of egg distributions

You cannot get any further evolution stages of pokemon from an egg, nor can you get a legendary from an egg.

These distances were minimum distances at the start, however, that appears to have been fixed

I've found an updated version of what you get for every 2k, 5k, 7k and 10k egg.

You can also view the updated eggs on the Silph Road here

I plan to keep this answer updated when the eggs change.

Feb 1 2019

enter image description here

Jan 16 2019

enter image description here

Update

The eggs change so frequently that it's difficult to keep up with replacing the answer here. So please refer to the Silph Road for the updated list.