Your water is eating your copper pipes because of its acidity, not its hardness. You need an acid neutralizer.

As for the sheets, I would first try the longest, hottest wash with oxygen bleach. If that doesn't work, try chlorine bleach.

When you fill your sink with water, does it take on a blue tint? Hard water is not the culprit. In fact, hard water is your friend because it helps to coat the interior of your pipes with scale and minimizes water's contact with copper.

Soft water sometimes causes this issue, but since you are on a well I might take a look at your electrical system believe it or not. I lived in a subdivision in California about 20 years ago and everyone had "blue water". It took over a year for the water company to determine that electricity was leaking into the copper pipes via some ground wire that they use to keep the mains grounded. This caused a cathodic reaction with the copper leaching into the charged water.

You might use a chelating agent similar to CLR for your stain. I think the citric acid type might work and it is pretty cheap. One at Walmart is LemiShine. It is a rinse aid for dishwashers but you could try to just dump most of the container in your washer and use the hottest water for the longest cycle. On mine it is a sanitize cycle. Set to heavy soil and extra wash mine runs slightly over 3 hours.

Thank you for your replies. It makes sense that this is acidity.

Sshrivastava - definitely water in the sink/john has a turquoise tint. No clues on the electricity thing, though.

Nerdyshopper - I am so not a chemist - does copper form a chelate with whatever is in LemiShine? Is that a similar process that I got when I used "Iron Out" to remove rust stains from white clothes? Which, by the way, was like magic!

As an aside - I don't drink my well water, so my internal pipes aren't copper coated, I hope...

Copper is unique but it is a metallic ion and should be bitten by a strong chelater. Not sure citric acid is strong enough but it can't hurt to try (plenty cheap). A stronger one is oxalic acid which I used to get at a paint store, but it is poisonous and might damage your washer. Since Lemishine is used in dishwashers, I doubt if it would hurt your washer. Or you could just put it in a dishpan with hot water and dip a corner into it for a while. The color will tell you if it is working.
I'm a retired chemist, but I am 80 years old and it has been a long time since I plied my trade.

Nerdyshopper - I appreciate the voice of experience! I think I will try the dishpan method.

If it doesn't work - it's really a kind of pretty turquoise ;o)