No, at the expense of the US tax payer, who paid for the credit card processing fee and free shipping through the US mint.

More accurately, at the expense of us currency holders. The mint is self-finaning.

The cost to the mint to produce a coin is far less than the face value.

Whether anyone was actually hurt depends on how the cost to distribute this way compares to the normal cost to distribute new currency.

And, assuming this method was more expensive, calculating who was hurt and by how much would take me a few hours. Maybe someone else has that more on the top of their head.

Those airline miles are paid for with the processing fees charged to merchants by credit card companies.