What insulation should I use in a stone basement?

Q: What about solid stone walls/floors and insulation? Ensuring breathability.

A: Probably the best solution here is to use a product like rockwool boards for insulation. They are breathable so you reduce the risk of mold behind the boards. They are also fire proof, unlike most foam insulation. It is probably a good idea to also install a heat-pump hot water tank. Unless you have very high electricity prices these are cheaper to run than heating water with a boiler, even one powered by natural gas. If you use solar panels to generate your electricity it will be much cheaper than heating your hot water with natural gas. They also dehumidify your basement which will reduce the risk of mold.