I think you're referring to short circuiting the battery, which can lead to wires melting their insulation and setting fire to nearby items, not earthing to the battery. And no, that is not your problem with melting grills.
The head unit should be earthed to the chassis, the fact the earth "couldn't be found" shows that the installer isn't any good and has no skyline specific knowledge, perhaps take your head unit to somewhere reputable for installation.
I would guess that 44c+, direct sunlight and trapped in a closed up car is pretty damn hot, although I've never seen the grills melt before. You should also watch out for coil/spider/surround glue melting, might think about screwing in your speakers properly too, how can that not be vibrating and annoy the shit out of you?