Modern Buddy

What lights are supposed to come on in the speedo when I turn on the key, but before the scooter is started?

Gas Gauge?
General instrument illumination?

Brake comes on if I pull it.
Neutral light is out - but i think i know what's the problem there.

Also, a more general question, does the engine charge the battery? Or is this why everyone uses trickle chargers?

Hal

Engine charges the battery. Here in U.K. nothing comes on till the engine fires up. But I presume your wiring different re brake lights and a few other things we don't even have a neutral light. Clutch also needs to be pulled in on my 2t before starting.
Trickle charge is for when the bikes off the road for prolonged period i.e. winter to stop battery discharge. I leave mine up to 6 weeks in a unheated garage and never get battery problem but depends on how low your temperature goes down to don't take many sub zero nights to kill off batteries