Am I correct in assuming your problem is with your wired ethernet adapter (NIC)?

Yes, the highlighted one is the real-one (and only one that really matters ... the rest of those are more software-related).

So, when it disappears ... is there instead a new "Unknown Device"?

Have you tried re-seating it (or is it on the motherboard)?

I have seen gigabyte LAN not present in Windows dev mgr because it got disabled in bios of an old optiplex model. 
Given the age of 8300,
Try replace cmos battery with a new battery. also clear cmos settings via motherboard jumper. If issue still not resolved, try a clean os install.

No - nothing appears as an unknown device.
I tried to reseat it on the weekend, but it looks like it's actually part of the motherboard.
It's still happening.

Tried a new battery on the weekend. 
Didn't fix it.
Any good way to keep all files and installed programs with a clean os?