Wow, that's some incredibly bad advice.
Yes you can go from Lightroom 4 to Lightroom 6.
As far as crashing goes, when the computer shuts down and rebotts, 99.999% of the time you have a hardware problem or a driver incompatibility. The problem is not Lightroom. You need to update your video driver (or sometimes downgrade your video driver) and check for hardware malfunctions. UYou say its not overheating, but fail to give your reasoning. Lightroom puts stresses on your hardware that other software does not, including some games (again, you give no details aabout what games you are running, although I wouldn't have a clue if you named names how that game stresses the CPU vs GPU vs memory vs hard disk).
(Not sure why this is getting rejected by moderators, it's a followup to my question)
My reasoning for ruling out overheating is that after sitting overnight it dies withing 5-10 seconds of me double-clicking the Lightroom icon (if I've already tried to import). That's way too quick for any overheating to happen. I have an aftermarket cooler on the CPU that's capable of handling plenty of overclocking (which I haven't done any yet though). But I agree, the symptoms do scream of hardware issues, but since Lightroom CC (not Lightroom 5) is the only one showing it, I thought it was something Lightroom specific. I play lots of hardware taxing games, which is why I got myself such a beefy machine primarily and I feel like they combined would tax it pretty well. I also felt like I could rule out GPU overheating/power/drivers by disabling the GPU boost component in Lightroom.
However, to be thorough I downloaded Prime95 (a CPU and memory "torture tool") just to make sure, and it turns out to be triggering the problem as well. So that rules out something Lightroom specific. I'm just surprised I haven't encountered this before. I'm going to run a comprehensive memtest86 test to see what that turns up, but I can't find myself a USB stick at the moment so that will have to wait. If you have any advice for other diagnostics to run to try and isolate what the problem is that would be greatly appreciated.