Backlight bleed is hard to evaluate via a picture, but yes, that's probably pretty normal. The solution tends to be not to look at black images in a dark room.
From a late '13 MBPr I suspect you'd only get a 4k resolution, and no, I'm not the least bit surprised that it struggles. Protip: Unless you're planning to upgrade your computer soon it would be better to get a 2560x1440 screen similar to the now retired Thunderbolt Display. You can get a nice IPS screen at that resolution and with no more backlight bleed pretty cheap nowadays.