I've solved the problem with the help of some experts in another forum. I wanted to leave this answer here in case it helps any of you with your random crashes.
This litany of crashes was, in fact, caused by incorrect memory timings in the PC's BIOS. The big tip off happened after we scanned memory for the Nth time, it came back with no issues, and then we investigated the Windows Event Logs. We found that even standard system services and user applications were crashing (the SuperFetch service, for instance) at random intervals.
The solution was resetting the BIOS to factory settings (had to adjust a jumper on the motherboard to clear settings). Once the BIOS reset was complete, the system rechecked the memory timings automatically and loaded up the appropriate defaults.
I can only assume the timings were off because the system used to have 2 sticks of RAM from another mfg, and then I adjusted the BIOS to overclock the CPU and memory. I removed the original 2 sticks and installed 4 brand-new ones without checking the memory timings or BIOS settings.
If you're interested in following along with the steps that we took to get to the bottom of this issue, you can read all about it here:
http://www.overclock.net/t/1339557/a-year-of-bsods-mostly-ntoskrnl-exe
Thanks to all for the suggestions!
Best Answer
Winaero posted a guide how to do this. You need to download the updates KB2990941, KB3087873, the NVMe driver from samsung and integrate them via DISM into your install.wim/boot.wim: