Black screen of death persists, I am out of ideas..

Hi!

So, as the title suggests, my laptop (HP Probook 4525s) keeps on torturing me with the infamous black screen of death.
It started about 8 months ago, as far as I remember, the Windows froze, I forced a shut down, booted again, and after the "Starting WIndows" animation, it failed to boot up completely. Black screen, just the backlight of the screen. I started searching for solutions, tried a lot of Catalysts (the first thing they suggest when this problem appears is to update Catalyst), to no avail. After a while I gave up, and started using it without VGA driver (no sleep mode, no Aero, etc). Then I decided to try 8.1.. even after installing a new op. system, the problem didn't disappear. Then it hit me home, it's a hardware problem. Then, somehow, the problem solved itself. Windows managed to boot, sleep mode, everything right.
I was overly happy. It worked well for months, until about last December, when it started again.
I wasn't really happ with 8.1 (for numerous reasons), so I decided to downgrade to 7. I did it, then install the newest Catalyst driver, and it worked well again..

However, a few days ago, it started BSOD again. In the last few days, I tried a lot of Catalyst versions, again, to no avail.
I gave up again. I deleted Catalyst with DDU, and again using it without VGA driver.
It boots without problems, however, no sleep mode, no Areo, few other functions are missing too, and my GPU temp is constantly between 75-85 degrees (Celsius).

My question is: is it a HARDWARE or SOFTWARE problem?
Facts: if I delete Catalyts, it boots without problems (also in Safe mode).
It managed to boot with one certain version of Catalyst (i don't even remember which one), but Windows Experience Index didn't complete, gave 1.0 for graphics, and the GPU temp remains at 75-85.
I spoke to some IT guy in Sept 2015, he told me my laptop need BGA reballing. What about that?

tl;dr: black screen of death, tried many versions of Catalyst to no avail, even tried 8.1, worked well for a few monhts then the problem came back, now I'm on 7 again, problem persist, it manages to boot with one certain version of Catalyst, but WEI doesn't complete, gives 1.0 for graphics, GPU keeps on boiling, still cannot use sleep mode and Areo; boots up without problem if there is no Catalyst
Hardware of software problem? Would BGA reballing help?

Ati Radeon HD5470

I am not a PC user, so it's not like I can install any VGA I want

Also, I don't play games at all, so I don't even need a dedicated video card. I bought this laptop 5 years ago (when my IT knowledge was literally as little as zero).
My next computer (either PC or laptop) won't have a dedicated video card. A simple Intel HD 4xxx will do it.

Fortunately, I wrote about this problem on other forums as well, and came to conclusion that it's a hardware problem, it needs a VGA chipset reballing.