Grey strip on the screen

I've got good news and bad news ...

The good is this is something you can fix!

The bad, it's going to be expensive as the display its self is damaged.

What likely happened is you either banged the system or something heavy sat on the lid damaging the display. Unlike older displays the newer systems have thinner screens so they bend vs crack. In this case the connections on the back of the LCD panel have come loose (often called a tab error) so the driver logic and the connection onto the main logic board is damaged.

Here's the IFIXIT guide for your system: MacBook Pro (15 Zoll, Ende 2013, Retina Display) Displayeinheit austauschen. The part is about $500 U.S.

You may want to review this IFIXIT teardown: MacBook Pro Retina Display Teardown. Replacing the cable is a big job as well. But given your symptoms I wouldn't suspect it.

Unlike the older LVDS signaling the signaling the display uses here doesn't distort with stripes or other colorful artifacts when the cable is damaged. The issue is now no signal or dropped signal if the cable is damaged. Unlike the older non Retina models I haven't see any damaged cables without it being a bang or liquid damage as Apple improved how the cable is routed.

Given the new info that you encountered the issue during heavy game play when the system got over heated I'm wondering if you have a graphics logic issue. In this model the CPU has the graphics logic would would mean the logic board is bad. But, that can't explain the movement of the screen cutting the image in & out (grey bar) that is the symptom of a damaged display.