Water Distillation Project
Obviously in Australia, we have access to much better quality materials and tools, and you can purchase activated charcoal rather than making it yourself. It's the idea created by "Prof. Josh Kern" that makes it a great off-grid project 🙂
There are more video's and information at: charcoal-biochar-water-treatment
You've already made a good start with the system engineering you've done so I will only offer some comments:
1/ re batteries. I would use two batteries, one lightly loaded and hopefully more reliable feeding the electronics and another independent one for power components. This way a discharged battery with near zero volts will not disable the electronics needed to charge it. The battery used for electronics could be on a constant float charge fed by a small solar panel and would last forever. I used this for my boat battery for years and it worked very well. (and I expect we get a lot less sunlight than you do.)
The circuit diagram you've picked up for battery charging uses an Arduino circuit board for charge control but there is no mention of software for the Arduino. For a lead-acid battery a commonly used algorithm is: charge at constant current until the voltage rises to a preset level then charge at constant voltage until the current falls to a preset level then change to a constant low level current for the float charge. Other batteries such as LiPo will require a different algorithm.
2/ re solenoid valve. A DIY job sounds like an invitation to pesky problems and fiddling. Take a look at Adafruit product code 996 and 997 for a ready to go 12v solenoid valve which I hope is in the size range you need.
3/ re borehole water. Where I live deep well water water is normally quite hard. OK for drinking and cooking but requires a lot of soap for washing. But it quickly leaves a deposit of lime in any plumbing carrying it so if yours is similar you may have to think about an acid flush. Your mention "flecks of iron" makes me think you well water is also quite hard and the flecks you see are precipitating out when the pressure falls.
That's enough for the moment but I'll probably think of more to expound about later.
4/ re plastic. I've found that plastics are sensitive to sunlight in spite of the "last forever" mythology. The transparent acrylics I've used, plexiglass, perspex tend to develop crazing (tiny surface cracks) after a few years use in sunlight. Transparent polycarbonate (lexan) develops a frosted surface after a few years in sunlight. Both still pass light but it is diffused and (I expect) weakened in the transmission.
5/ re collecting roof rainwater. I hope you don't have a lot of birds roosting on your roof.