Interesting pictures. The low KH is causing problems with rapid pH swings. How high is your GH?
I prefer not to use stabilizers but I suppose you could. A phosphate buffer might work but try running the system without this for now.
Short of replacing some of the water with RO or distilled water I think your probably stuck with the high GH. It's the ions that provide the plants with their nutrients - most of the methods to remove ions take the ones you don't want and the ones that you do. Plants will remove some of these ions but you'll be adding some back in by adjusting your pH back up or when you add topup water from the original source or even fish food.
I'm not sure about neons but I suspect that for most fish the GH won't be a problem and I'd probably raise the pH (and KH) using potassium bicarbonate and then maintain them using a mesh bag filled with crushed coral which is calcium carbonate. You'll still raise the GH because both add a mineral ion along with their carbonate ion. Usually the pH will gradually decline and you'll have to repeat the potassium bicarbonate treatment but you should watch the pH in case it goes up as well. If it does, remove the mesh bag with crushed coral and allow nitrification to bring the pH back down. Once this happens put the bag back in the system.
Nitrification can shut down when the pH drops too low so get the pH adjustment done ASAP but do it carefully, it's easy to overshoot.
Swirl filters are designed to catch settleable solids only and won't do anything for suspended solids but will at least help reduce any accumulation in the grow beds. It's OK that the suspended solids make it into the grow beds. You actually want some of the solids breaking down in the grow beds because it provides more nutrients for the plants.
Springtails are detritovores and eat dead or decaying material - probably won't hurt anything but can be annoying. I've found that these are less of a problem with outdoor systems. If you want, you can reduce their numbers by flooding the grow beds over the top of the media after removing the bell on your siphon so that they can just float over to the standpipe and out - this isn't going to get them all but might help a bit and hopefully the fish will eat the ones that flush out.
The wetted surfaces of perlite and rocks in the grow beds will be providing habitat for bacteria that break down the ammonia into nitrites then into nitrates which the plants use. Assuming your system is cycled then this is already happening. If it's not cycled or if their is a problem with the bacteria in the grow beds then ammonia will build up. If you're getting an ammonia build up check to see if it's at toxic levels for the fish and then stop feeding if it is. A table for ammonia toxicity is here -
http://ibcofaquaponics.com/information/The phosphates will be taken up by plants or algae (preferably the plants) and used for compounds like ATP which is involved in energy transfer. Bacteria developing on the surfaces will also use this up as they grow and multiply.
The toxin locking and water priming supplies aren't used much in AP - probably isn't much harm to them but most are using their system as part of their food supply so we err on the side of caution. Whether the water priming supplies work really depends on what's in them - sometimes it's not the right bacteria or the bacteria are no longer viable because they've been held on the shelf too long. The bacteria that perform the nitrification process are so common that eventually your system will wind up with bacteria adapted to your environment regardless of what you add (either the added bacteria will adapt or the right ones will arrive and take over). Some of the toxin locking products can affect the readings from your test kits.
You should take a look at the wetted area of your growbeds in order to figure the filtration capacity and how many fish you can care for with your system. I'm thinking that you may be overstocked but I haven't looked it over and I could be wrong - I'll let someone else chime in on this. Usually it's based on the final size of the fish. Here's a link that might help but there's more than one way to figure it out -
http://www.backyardaquaponics.com/forum/viewtopic.php?f=12&t=6646FYI - Most people here use the API Freshwater Master Test Kit for doing their tests. You may want to see if this will give you the tests you need. If you've already got something else ordered, not a problem
PS - You can post instalinks but we prefer that you post directly to the forum since pictures get lost when outside sources change or start charging
. You can find a couple of threads on how to do this near the top here -
http://www.backyardaquaponics.com/forum/viewforum.php?f=4
Backyard Aquaponics
(LOWER GH - RAISE KH - RAISE AND SUSTAIN PH) 
. The air and water are free out there but the Internet services will cost you and ... help you if you need to find some civilized facilities. It's the other way in the city. Hope you're having fun on your hike 