An update: both Windows Memory Test and memtest86+ revealed memory errors. I took a chance a purchased new memory even though the problem could be the motherboard etc. The memory was quote expensive now that it's basically legacy memory (DDR2).
When I went to replace the 4 x 2GB RAM sticks, I noticed that one set was generic and the other set was Corsair and the same model of RAM I just bought. So I replaced the generic RAM sticks with new Corsair ones and re-ran memtest86+ and it still found errors. I replaced the old Corsair RAM sticks with new Corsair RAM sticks and was able to survive a memtest86+ test.
The system is now running Windows 10 and we'll see if any more BSODs occur. I'm disappointed that I had to replace the old original Corsair RAM with exactly the same RAM, just new!