Yes, if you have a 12 volt battery and/or a charging system to support it.
Hook a 12 volt light to a 24 volt system and the light will burn out in seconds. You need a step-down voltage converter to do this.
Never
No, you can not hook up a 12-110 volt inverter to a house socket.
in series
Separate the 2 12 volt batteries and charge them individually.
Yes
To recharge a battery you put the solar panel in the light and hook the red cable on the positive and the black on the negative
No Yes. hook two in series, two in parallel.
Hook then up in Series. Positive to negative. You then have 24 volts but the same amperage of one individual battery.that will short out other battery
You must hook them up in series. You will need four 12 volt batteries to do this.
No one answer is correct as each scooter may be different. On my scooter I have a box built into the floorboard with a waterproof cover held in place with two screws. Below the cover is the sealed lead/acid cell and the fuses for the scooter along with spare fuses.
If you want your bulbs to last longer then use 130 volt bulb on a 120 volt system. The light emitted by a 130 volt bulb, when comparing it to a 120 volt bulb, will be a bit dimmer.