I agree with Old.Tony, this has all the earmarks of an earthing issue, which for most vehicles is the negative side. That's always the place to look when multiple electrical systems, which (should) have no direct connections between them, start to muck up at the same time.
That "resetting" issue is puzzling but for now, I'd ignore it. It's probably like the error codes some people get from their vehicles. It might throw up 10 apparently unrelated errors but when you fix the correct thing they all disappear. Over the years I've seen some really odd behaviour in vehicles because of crap earth connections. Just focus on finding and fixing that bad earth and see if you still have issues afterwards.
Early Ford Falcons and, to a slightly lesser extent, the Holdens of the same period, were notorious for have oddly behaving rear lights. It was due to the practice, which was common at the time, of relying only on the screws that attached the light assemblies to the body to provide the earth connection. That was fine when the cars were new but once the rust settled in, as it inevitably did, those connections were no longer up to the job. You'll still see it very frequently on trailers, especially older or homemade ones, where some people simply rely on just the coupling or the trailer metalwork to provide the earth connection instead of properly wiring it into the plug and socket and at the lights themselves.
