Won't stay running when engines warmed up
 

Just for side info I recently did a head gasket and timing belt on this 2.0L. It starts and runs just fine when cold but when you shut it off and restart it when engine is warmed up it will not stay running .also once it start's cold and warming up it starts to idle rougher and rougher. Another issue with this car is when you take off the oil cap a white haze comes out pretty heavy but I don't think it's smoke.

Have you performed a compression test?

Depends on what you mean by 'heavy haze'. Noticeably to much haze can be a bad thing. Haze after a warmed up engine on shut down is normal in most cases. Lets get the cheap repairs out the way first.

Check for a check engine light. Also do it have slight/spotted misfire while driving? All can be caused by a/both faulty ignition coils.