How does egg hatching distance relate to the hatched Pokémon?

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

2 KILOMETERS

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

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Jan 16 2019

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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.