Your GC goofed so its on him to fix to your satisfaction. That probably means removing that tile, setting a new one and making a round hole. On his dime

Sounds like the easiest solution would be to get a bigger light.

"The round back plate of the light doesn't completely cover the corners of the blue box."

How much are you missing it by please? If not too much, you could take a belt sander to the errant corners without losing the tab stop function. Or lose the tabs and mount the box with Madison Hangers.

Thats just an old work cut in box. Replace with a round cut in and mud around it (make the GC do this) Black is the only UL listed to hang a light, but youll want to check the weight rating and verify the light is within that spec.

This is the classic GC non electrician doing the work. Doesnt know what kind of box to use and doesnt even bother to strip the jacket back on that wire.

i wouldnt let that guy do anymore electrical work after this.

Is there anything prohibiting them from removing that piece of tile, correcting the junction box, and then installing a new piece of tile? This is what I'd recommend unless there's something serious in the way that would stop the tile from being removed. This should be at their cost.