Wow. 20Gb is a huge amount if stuff to get rid of, generally you don't find more than 3-4Gb in either of my pc's, I use CCleaner at least every other week just to get rid of all the internet crud my daughter seems to accumulate.
65°C while gaming is perfectly acceptable for a cpu and stock cooler, and age of cpu is irrelevant, cpus themselves are the single hardest component in any pc to kill. The whole pc will get old and die or get out-dated long before a cpu calls it quits. I have a 20 year old pentium II 350MHz (OC to 400MHz) that still works like brand new.
Hdd possible. It's a couple of spinning disc's, spun by black rubber bands. Yes it's possible for that to get corrupted, or if you picked up any malware, get corrupted, or even a slight bump while it's working. Generally, figure on @5 years healthy life for a hdd, after that is usually when you start to get bad sectors. Easiest way to check is with WD software, smart checks, from a Windows Admin CMD run CHKDSK which will also look for errors.
For ram, use memtest86 and run it overnight, it's by far the best ram checker.
Basically, Warframe is telling you exactly what I said.
You have a 4th generation Intel cpu and motherboard, so the motherboard drivers for Lan and audio are much older code-wise than Windows 10. Since Win10 came out, it has had some few issues with what it considers 'Legacy' drivers, ones not designed to work well with a 64bit OS, since it wasn't around. Win 10,however, is stubborn to a fault and prefers its own drivers over others, resulting in conflicts between the 2,and you get a bsod as a result. It's very common, and nvidia has had many complaints about this, even today there's still issues with some games and even cards like the gtx1060 not working right simply because Win10 is messing things up. So if running hdd, ram tests etc reveals nothing, I'd still recommend you upgrade your bios and motherboard drivers from the HP website, and if they have none, find out who makes that mobo (usually Asus, msi or gigabyte) and see if they have some.
If all of that resolves nothing, when you purchase the new hdd, use that as a brand new installation of windows, and just transfer all your game files from the old hdd, format it, and keep it as a spare for long term backups etc that almost never get used.