Quick follow up:
I've run the Intel diagnostic tool and passed everything except the base clock test, which appears to be because the tool has not been updated to support 2nd gen i5/i7 processors. The intel burn test didn't phase my desktop on maximum settings in the slightest.
I'm starting to think this is a problem with something other than the CPU, although I'm no expert.
You're right about the Processor Diagnostic Tool, the base clock seems to be hard-coded to 133MHz, vs Sandy Bridge's 100MHz.
Your other problem might be a missing or old driver problem, it doesn't seem to be a CPU problem.