Sounds like a HW problem, but we presume you attempted system restore or OS refresh (if you are backed up and not worried about losing data).
1)Did the problems start after a specific event (Updating BIOS)?
2)Why did you update BIOS?
3)It might be a heat problem. I recommend installing heatsink software to monitor CPU/GPU. Many of these software packages can contain malware, but I found a QUALITY open source package (Michael Moller-Open hardware monitor) <---search for it. If you're running hot, you will need to reapply thermal paste (Arctic Ice etc) to your CPU or GPUs. Please visit YT for proper procedures (many vids). You say you reballed the GPU- make sure you are doing it correctly. Do not use the credit-card technique. Single small dot in center and compress (no scraping!!!). Many YT vids show proper application. But first install the heatsink monitor software.
3a)As diagnostic for 3), is your fan on constantly or cycling more than usual - that's obviously a huge tell.
4)I also recommend YT vids for taking your laptop apart. I searched and found your model. I didn't watch all of it to ensure quality, but many of these vids exist and can save a lot of heartache. Here is yours:
watch?v=FMK8SOYLxck
Good luck
Tom D.
Siva S.
01/10/14