I don't know much about the 250s, but since no else is chiming in...
If it has a CDI, it should be 12v. If the original lighting coils output AC which I have seen on snowmobiles I had, then you would need an AC to DC converter to get the DC power to charge the battery. When you measure 9V, what RPM is that at?
If the solenoid has 4 terminals, 2 big and 2 small: The starter would connect to one big one and the other to positive on the battery. The small terminals would be connected to ground and the starter switch. The other side of the starter switch would connect to the positive side of the battery. It could also go positive, solenoid, switch ground. Basically you are connecting positive and negative to the solenoid interrupting it with the switch anywhere along the way.