Need expert for difficult computer problem - random crash/lockups

That OCCT is a neat tool. I'll have to run the tests a bit longer, but 10 minutes with each test so far and everything looks great. My mobo/battery theory is still alive, for now.

10 minutes may not be long enough to flesh out any issues with the CPU or RAM. You might need to let it run for several hours. Make sure to keep an eye on your CPU temps as well. I'm not sure how intensive the testing is in that tool and if you don't have proper cooling for this kind of workload you might get a false freeze/crash/reboot from the overheating CPU. If it runs for several hours with no sweat and it's completed at least 3 passes of memtest without any errors, I'd say move on to testing the hard drive. You can use a utility like PassMark DiskCheckup - SMART hard drive monitoring utility to check the SMART status of your drive(s) and run tests on them to make sure it's healthy.

Also, you should fill out your system specs in your profile so we can know exactly what we're working with, hardware-wise.