RE the gearbox issue - perhaps it will be obvious once opened up, but given there's a significant gap in my knowledge of such things, I thought I'd query the MM brain trust here now that I've ridden it and can provide a much clearer description of the issue.
In short - it's near impossible to keep the bike in third gear. It shifts into it, and if you baby the throttle and are essentially putting zero load on things, it will stay there long enough to shift to 4th. Mostly though, you need to shift from 2nd to 4th right away as 3rd is useless.
It does not permanently pop out, i.e. into a false neutral. Rather, it will rapidly go in/out of third gear every couple of seconds. I only did this a couple of times, not wanting to cause further damage - but perhaps that description narrows down potential issues?
Thanks in advance. I'll document the repair and eventual restoration as it take place.