My #2 350 is acting much the same way that your is. I thought that the carb was dirty on mine, so I cleaned it, but it seems to have made no difference. Also like yours, it will only idle for 30 sec. to a minute or so without some throttle input by me. And like yours mine will re-start after a 2-4 kicks (my battery is dying so I have to resort to the kick method). I was going to try swapping cdi's from one bike to another, but I have not gotten around to it yet.
As far as the lighting, you may have a bad lighting coil, but assuming that your bulb is good (I paid $2 for a new one from the dealer), check for ground in the connection where the harness plugs into the bulb itself. I ran an aux. ground and low and behold, I have a working headlight ( I still have to aim it properly though).