It is usually good to get a lab report on your raw water. I like kit 90 from
http://www.karlabs.com/watertestkit/
What you have may be useful. You might want to clean the controllers and maybe replace the media. It does seem unusual to have that backwashing filter following the softener.
Looking around, I see the Watershield Gold can have various media and may even have a mixed media. It uses a Clack controller. I think that controller will have plenty of capacity and it should not cause too much backpresure. Your media could be Centaur Carbon. My iron+sulfur backwashing filter uses Centaur Carbon. It is very effective for me, but my iron+sulfur levels may be less than yours. Mine uses a chlorine solution during backwash, and your system uses air. The BLFC controls the backwash rate. Mine uses 5 GPM for a 10x54 tank. Other media use higher rates. If you were to put in a new media, you might consider Katalox Light. That would probably be good with 10 GPM for a 10 inch tank in Florida for KL. Your filter might have Greensand media, or some other media.
You might be well off to have a local dealer who handles that product tune it up for you.
Some mechanical Fleck 5600 controllers are timed, regenerating every x days. Some are the Economizer type where a meter causes the softener to regenerate after some number of gallons. If yours is the economizer, you might want to consider cleaning it up, and maybe replacing, or at least cleaning, the resin.
I am pretty sure the filter should go before the softener. So it is likely that was not installed by a professional. The lawn water should normally not go through the filter, and for sure it should not go through the softener.
For your outdoor system, you need to protect against UV. The tanks need a covering or be painted. The PVC pipes need to be covered or painted. It looks like your pipes have been painted, but you need to keep that up.