1) I never had a problem with mine like this but it'd be worth disconnecting the gauge and checking what voltage you have from the level sensor. E.g. empty the tank and see if it varies when you pour in fuel from a jerry can, basically to decide if it's a sensor or gauge fault.
2) The Superlight runs "Harley-Style" push button indicators. I'd suspect the RH switch is faulty. Find the connector for the right hand cluster and check the switch is actually working after that I'd suspect the relay. (Should be blue/red wire for the left switch, red/yellow for the right and black for ground.)
3) For the rev counter I'd suspect the mechanism has some grit in it or something. If you disconnect the tacho cable from the engine side you can try giving it a whizz with an electric drill and see if it does the same thing.
4) These Chinese gearboxes are a bit shit

I often had a game of hunt the neutral (seem to be affected by engine temperature and oil quality) but not finding 1st doesn't sound good

Might be the gear selector drum is worn but you'd have to dive into the engine for a peek at that.
5) I had the EFI version so I only had the issue that (deliberately for emissions) it was always running a tad lean. From the manual the carb looks like the most generic diaphragm-type one could get

This is less a Keeway specific issue and more of a shit Chinese carb thing. Usual stuff: check diaphragm for tears, float height, etc. If you're sure the carb's nice and clean you could try moving the clip on the main jet needle down one position - should add some fuel in the midrange.
I'll PM you the service manuals. Badly written and badly translated Chinglish but better than nothing
