here's my thoughts:
not idling-you did pull the jet out to clean it, right? see daylight through the orifices and down the center of the pilot jet?
not restarting after warm-have you checked the valve clearances?
does the float bowl have gasoline in it when it dies? just wondering if it is burning all of the fuel in the bowl, dies, and the needle/seat is stuck and won't open up to meet the fuel demand until time has passed and then it has fuel again.
another thought is this: is the fuel tank vented so it won't reach equilibrium and pull a vacuum on it thus stopping the flow of fuel to the carb? just asking cause i'm not real familiar with your quad and i don't know if it has a fuel pump or not. just trying to help.