I don't understand why the dirty water, shutting a pump off and turning it back on after draining some water from a water heater or the water line to it shouldn't cause dirty water. Are you sure you don't have a leaking check or foot valve in the well?
It would explain the stuff in the control valve and it running ir backwash, because a leak would cause a siphon and suck water backwards through the softener.
I'm sure you can take your water powered valve apart, I'm not sure you can get it back together without problems but, taking it apart without knowing what is wrong with it if anything, probably isn't a great idea. And if you're not one of those feminized American males you put a piece of duct tape over that bubble so she who you love and admire and must obey doesn't see the boogers you'll probably forget all about it in a few days, or a week.