Your pressure tank is in the wrong spot. It should be after the point of injection. I'm not convinced that its the source of your problems, but its likely not helping.
When the carbon media is new is contains a large amount of dust in it and unless it is well rinsed it will make its way into your water. I would manually regenerate both the carbon filter and the water softener a couple of times in a row to see if you can clear up the water.
My question to you is do you know if your carbon filter has been properly sized to the GPM output of your well? If the carbon filter is oversized it will not properly backwash.
Did anyone mention air injection systems to you? Just wondering because I also have H2S problems (mine isn't as bad as yours) and that is how I deal with it.