I had this exact issue with mine too: occasional green flicker when the screen was off (AoD), random streaks, later on also the screen turning completely off, all exacerbated by heat. In my experience I had to go to an EE store (UK) which is listed as an authorised repair centre. They apparently use a company called SBE for the actual repair and they tried to claim fractures on the digitiser (couldn't get anybody to show me any proof of this) and quoted me almost £200 to repair it. Instead I took it back and had to just RMA the whole device. This was all despite the phone being in great condition otherwise with no drops or dents or anything. Wasted several days on that whole thing, all told. One day wasted handing it in at the EE store and getting absolutely nowhere and overall I reckon if I'd been able to arrange an RMA on the same day I would have saved an entire week.
The EE people in-store seemed nice enough but the people they forward it to seem really dodgy based on that. Despite confirmation it was under warranty and going to a Google Authorised Repair centre, SBE Ltd tried to charge me for it. I don't see what's to be gained from that, as if it's confirmed to be a warranty repair I imagine they would communicate with Google if they needed to be reimbursed, so I feel like the only reason to charge me would be to try and double dip.
(I got a text a few days afterwards asking me to rate my experience. I would have rated it a hard zero but my text wouldn't go through. I think because I'm not on EE's network it couldn't communicate to the short code properly.)