Here is a prior question with answer that might help -
If ONLY the hot water comes out black, that eliminates the well, pump, and pump pressure/air tank and associated distribution piping as well as the water softener (unless it is only on hot water tank feed side) as sources - which narrows it down to water heater or hot water pipes.
Black water means something corroding or building up for a long time, not a short-term issue like a clog - again assuming cold water is clear and that the water softener is on the cold water line to the house, not just hot water feed side. Could be iron or manganese bacteria buildup in the pipe that broke free somewhere, dirtying the line - but I would guess water heater is the issue.
And definitely yes - not maintaining the soft water absolutely could contribute to or cause the problem - soft water is as important to pipe and water supply system health as it is to taste and helping this get clean readily.
I would turn off the heat on the water heater till looked at - because a lot of sputtering (again, assuming only on hot side so low water in well/contaminated well water not the problem) I would guess you had so much sediment in the bottom of the water heater that it has started boiling at the bottom (assuming gas heater here) and has started going anaerobic (so would be stinky hot water), creating gas bubbles in the heater which then cause the sputtering when you first run hot water. This sort of overheating can cause catastrophic water heater failure, so I would shut off till checked out by plumber.
I would call a Plumber and have him check your lines (cold inlet and hot) for buildup - could be your lines are sedimenting with stuff getting through your water softener because it has not been maintained (either well sediment or mineral buildup), or could just be the water heater being dirty and maybe going anaerobic. Depending on how bad it is, just thoroughly flushing all pipes with maximum flow rate, flushing and cleaning the water softener (and refilling with the treatment media), and draining and chemically treating the water heater then chlorinating and reflushing the system might solve it. Typically about $300-500 charge for all those combined - get an estimate before starting work. Might also involve 8-24 hours out-of-service for your house water during chlorination to kill the bacteria in the lines/tank, if that is needed.
BTW - when lines are being cleaned dishwasher and fridge water/icemaker and washer and similar water-consuming appliances should be isolated until the lines are cleaned out and disinfected, then they should be flushed out (run an empty cycle) because otherwise they can get contaminated/clogged with debris from the lines. Because nothing is coming out on washer hot cycle, its water line filter (either at inlet of water hose or at the washer end) is apparently plugged already and will have to be cleaned out.
Of course, if the pipes are so badly corroded that they are getting blocked with buildup (especially with steel/iron pipes, usually to a much lesser extent with copper, rarely significant with plastic) then it may be that some piping replacement will be needed.
IF water heater is in bad shape it may need replacement - commonly in the $1000 ballpark if not more than 50 gallon. A quick visual check inside the pipes at a couple of fittings in the hot lines will let him know if pipe corrosion is a problem or not, or if it is just a water heater issue. If water heater issue and it is fairly new (say less than half life old) flushing and treating it may solve its problem (he should be able to tell with a fiber optic scope or bright penlight looking inside after flushing), but if badly corroded inside or the crown plate (bottom) is corroded and flaking or has a heavy buildup baked on that will not flush out then it should be replaced or same issue may happen soon again.
He should also check temperature setting on water heater - if set too low (below about 120-125) it can grow corrosion-causing bacteria - plus that low a temp promotes bacteria growth that can be dangerous, including Legionaires Disease bacteria.
Above was all about hot water system - if this is happening on cold water system too, then I would be looking at a corroded pressure/air tank at pump or well water very dirty due to collapse in well, heavy bacterial growth in well, or well going dry so pumping a lot of sediment. Running water full force into a glass bowl and letting it settle out for an hour or so would show if that is the case. Be sure to run the faucet a bit before test to flush out any dirt that might have gotten into the faucet spout from dirty hot water.