I see no reason for fine mesh resin, standard mesh will do just fine.
Changing the valve to a metered design will save you water and salt and is certainly a worth while thing to do.
Your iron is over the threshold that I like to see going through your softener but, if you are vigilant about running Iron Out or
similar products through the system regularly you might get away with it. If you already have air injection then I suggest cleaning the equipment up and especially the piping from the injector to the tank and go ahead and put it back in service. You can install shutoffs and unions so the bit of pipe can be removed and easily cleaned when needed.