Re: Dying when turning headlight on: you have a bad connection at the battery or the battery is dead. The battery needs to provide enough voltage to fire the ignition until you get the revs up high enough for the alternator to keep up. The battery may have failed or your charging system may not be keeping it charged; however, since the charging system will run the bike when you bring the revs up, it sounds as if it is OK. Most likely the battery is low on water.
Neutral light: The switch is bad internally (it's down by the drive sprocket) or the wire is shorted to ground somewhere. The neutral switch works by grounding a wire from the bulb, so if the wire is shorted anywhere between the headlight and the switch, the light will stay on.