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As far as I can tell, every circulation strike U.S. coin is copper or alloyed with copper .... There must be a reason. It may be as simple as durability.
To a large part that is the reason. Also the coinage metals traditionally are fairly dense, their density and weight in ancient times indicating their value. (Most common metals are significantly less dense than copper, silver, or gold. The exception being lead which due to its softness is unsuited for coinage. But useful in making "hefty" counterfeits.)
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Nickels are 75% copper, but are much harder than other coins, I assume because of the nickel content.
Correct.
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Are they still 75% copper because nickel costs more?
They were originally 75% copper because at the time the alloy was developed nickel was more expensive (allowing the coins to still have an intrinsic value close to face value) and coins of higher nickel content would have been too hard to strike. Even with improved dies and better control of annealing the planchets reducing the copper content would still be very hard on dies today.
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After retiring copper nickel small cents, it would have been a simple matter to return to all copper like the copper large cents. Instead a bronze alloy of 95% copper was used. Once again, I am guessing the alloy made the coin harder and more durable than just copper.
Correct again, the bronze alloy was significantly more durable than pure copper would have been. (hence the reason the romans had bronze swords and not copper ones. It wasn't called the bronze age for nothing.)
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Finally, there's the clad coins: hard copper-nickel plating over a softer core of copper, with the copper, I assume saving cost over using more nickel.
The clad composition came about for several reasons, one of the main being as an anti-counterfeiting measure. But the copper core also helped the coin more closely match the electromagnetic signature of the silver coins. Today it does have the advantage of being cheaper than a solid 75/25 copper nickel alloy would be.
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why copper has been in all of them.
Copper is not a precious metal, but it is a fairly common "semi-precious" one. It has several physical properties than make it ideal for coinage. If it wasn't so common, and didn't corrode so readily, it would have been considered a precious metal as well. At least in ancient times.
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Or someone in the Government owns a lot of Copper mines.
That's how we got nickel in our coinage, political influence on behalf of the owner of the only nickel mine in the US