Should I paint copper pipes exposed to nature?
I would not paint the copper. If you ever need to repair it, paint would be one more problem on making a good leak free connection. If you want to cover the pipe, use foam. Remember, copper water supply’s are quite common direct burial, the sun and elements won’t hurt it. I suggest insulation, if in a location that the temps drop below 27°F.
(Yes water freezes at 32°, but it takes time, in most locations 27°F is where damage to pipes starts as it takes ~4 hours at that temp to freeze your pipe.)