Lets try this, download this free app: [http://www.coconut-flavour.com/coconutbattery/|CoconutBattery].
With the system plugged in with your MagSafe power adapter take a screenshot of the main screen then disconnect the MagSafe adapter and take a second after at least 10 mins of being unplugged. Post them here so we can see them.
I'm suspecting your battery is going here or the battery charging logic has a problem. The issue of the display winking out is related as the system is going into power save mode and the system will then go into deep sleep.