BSOD, always involves ntoskrnl.exe

I have an older (circa 2010-11) Gateway NV55C laptop that is becoming ever more unstable. It was the wifes, so when she gave up on it, it became my project computer. It was overheating and crashing if you tryed to do anything more intense than web browsing, and the fan sounded like it was going to blow. The CPU cores were always roasting hot on SpeedFan, so I figured when I opened it up to clean it out I'd drop in a new processor. I found a new i5-480M on eBay to replace the P-6100 it came with, which I figured was pretty safe as they share the same chipset and the 480m was the top of the line option for an NV55c at model release. Well, I cracked it open and lo and behold the fan was totally stuffed with cat fur and dust, probably was barely or not turning at all. I wiped everything down with alcohol wipes and hit what i couldn't reach with some compressed air, pulled the motherboard and swapped out for the new processor, put it all back together, crossed my fingers, and booted it up. It seemed like I was in the clear at first, booted right up, detected my new hardware, let me run an experience index, and the overheating problem is totally gone. Well it stayed stable for a couple of days, and I was really stoked, seemed like I'd breathed some life back into a machine that had been starting to show it's age in speed even without the overheating issues. Then it started freezing on me, not often, once per 4-6 hours of use, and seemingly at random. Then it started BSOD on about the same schedule. Well I'm reasonably savvy so after I made sure it wasn't out of date drivers or anything obvious like my security software, I went ahead and did a clean system install updated windows and my basic drivers and if anything it was worse than before, so I did it again, figured this time I'd do the drivers before the updates and test performance with the original driver sets from OEM (Gateway). Well unfortunately now I'm stuck, I can boot up just fine, but almost anything initiates varying blue screens "IRQL not less or equal" and "Page fault in a non paged area", being most common. I can't even launch the Windows Update control panel without BSOD. OK, longwinded, but there it is, any help would be greatly appreciated!

So after a little temperature monitoring it became evident that the i5 480m was rapidly shooting up into the 70 degree Celsius range and then bluescreening. I used a high quality thermal paste (Arctic Silver) applied in a thin even layer across the chip when installing it, so that was not the issue. My understanding is that this chip should indeed operate at up to 100 degrees Celsius, though that seems high to me. I've reinstalled the old P6100, and with that chip the cores are running in the 45 to 60 degree range and no more bluescreens, stable for several days now. Could the fault lie with a bad chip? I did buy it off ebay, supposedly new and from a reputable seller, but who really knows? Could my motherboard (previously exposed to overheating before the fan was replaced) no longer be able to take the strain of a faster multithreading processor with higher heat outputs? I've uploaded the files you asked for, although with the system currently stable (even under the heaviest loads), I'm not sure if they will help. I really would like to boost this computer into at least the i3 range if not i5 range, but hate to buy another processor if this is going to happen again, what is your advice?

After doing some more digging I have confirmed that i series processors run much hotter than the dual core pentium (p6100), that i put back in my nv55c (still stable max recorded temp 60 at heavy load). My system instability and blue screens consistently started when SpeedFan and Speccy were reporting core temperatures beyond 68-70 degrees Celsius, which my p6100 never hits. I've ordered a replacement motherboard ($35 used from a high rated seller on ebay, sorry I am poor!), and if this doesn't fix my apparent thermal cap issues with the i5-480m, I was considering replacing the Gateway 1.3 (latest) BIOS, with the latest BIOS issued for the Acer Aspire 5742 which shares the exact same motherboard setup. Possibly there is an Intel factory BIOS for the HM55 Chipset that can be used? It is a risky step, but I figure at the worst I have to put back in the p6100 and old motherboard if I brick the 'new' one. I'd appreciate any input about whether these are feasible solutions, after all the i5-480m did pop up in my gateway BIOS, so I assume it knows how to manage it, right? Thanks everybody for your time and patience!