502/00 Charging Problem

I am having the provblem now that although the scooter starts easily and runs, it has odd behaviour of performance:
1. the battery is not charging so, as it runs, the battery power reduces. How can I test my scooter to find the problem? What can I then do to fix this?
2. The choke mechanism of the carburator only pulls up a small amount. The motor needs this to start. It seems to take a long time to warm up and run without the choke pulled. And if I stop the motor and restart moments later, with the motor still hot, it again requires the choke pulled to start. It there some sort of automatic choke mechanism working? Is there some sort of adjustment required?
3. Whether the engine is warm or cold, with choke pulled or not, when twisting the throttle the engine momentarily slows before picking up revolutions. Is this a timing issue at low revolutions? Is this a carburator adjustment?
4. I have not yet checked or adjusted the spark timing. Can you give me the spark moment before top of stroke so that I can check and adjust the points?
5. My warning lights do not work because the modern 12v 4w bulbs do not reach the bottom of the sockets. Is there another type of bulb I need that is slightly longer or is there another type of socket i can use?
6. Finally, I want to make the neutral light work but I am not sure how the switch works. When in neutral, does the wire from the neutral sensor send current or does it earth the wire?

Thank you in advance for any help you can provide. English documents are few and hard to find. Oh, and do any of you know where I can purchase a plate for the back of my Cezeta to mount a spare tire and the connector for my PAV 40?