That's why copper flashing is put on the peaks of roofs to prevent green growth just by having rain water wash across it and grab enough copper ions from the peak and wash down across the rest of the roof to kill whatever is on there (algae). I imagine putting copper in a compost pile will kill all kinds of stuff nearby and degrade the pile. Thermal conductivity probably isn't the problem (what's the conductivity of compost?). To have more hot water, you need more energy (temp * volume) so the pile just needs to be hotter or use a larger volume of water to store the temperature that is there. I think PEX is a good choice for tubing.
