I've only ever encountered one malfunctioning processor for sure. It was a Celeron 300a that I OC'd way back when. After a few years, it started to malfunction. There is a second one that may be in a Dell in our server room. It is throwing internal errors on one of the CPUs. Dell thinks it is a software issue though, which can cause it, and we need to rebuild the thing to check.
That's all I've encountered in 30ish years of playing with computers and about 16 years of professional systems administration.
In terms of thermal issues, I never worry about those these days. On user PCs the usage is light enough as to rarely, if ever, be a problem and if it is, the system will signal it. We get that occasionally, especially from the ultra SFF systems. On servers they live in a climate controlled room with high quality air filtration so no issues. Also their monitoring will automatically let me know if there's an issue. On my personal system I blast the dust once a month and have the CPU hooked to a radiator that can dissipate in excess of 600 watts, so there is more than ample overhead. :)