You have the Con score of your beast form.

The first benefit of the Durable feat states (PHB, p. 166):

Increase your Constitution score by +1, to a maximum of 20.

This is a one-time benefit you receive when you take the feat. You level up, select Durable, and your Con score goes up by 1.

For contrast, take a look at the other half of Durable:

When you roll a Hit Die to regain hit points, the minimum number of hit points you regain from the roll equals twice your Constitution modifier (minimum of 2).

This is an ongoing benefit that is triggered whenever you roll a hit die to regain hit points.

So Durable increases your Con score by 1, but when you enter Wild Shape:

Your game statistics are replaced by the statistics of the beast.

and your Con score becomes that of the beast form.

This does open the game up to potential shenanigans around leveling up while Wild Shaped, but that issue already existed, and this is one of the least problematic examples.

Yes, but only for a human, and only at the DM's allowance.

The only rules-based way to grab a feat at character creation is by using the "Variant Human Traits," found on PHB p.31. There you sacrifice four +1 to ability scores to gain a skill and a feat. It's not exactly sacrificing points from your point-buy, but the effect's the same.

Past versions have little bearing: they're different games, and while they're great to reference for source material, ideas, content-lineage, balance is the worst place to go looking at a different game.

From this edition we can see that your proposed homebrew solution would be rather unbalanced: it costs much less in your scheme than in the published version. (Which I presumptively consider balanced.)