Well, I think I figured out the issue. Right now, I've got almost everything installed (ALL of my apps, plus a couple of games) and am not having any issues.
What I *THINK* it was was the hard drive controller. The BIOS has options to set the hard drive in: ATA, AHCI, or RAID. When you set it to ATA like I did, the BIOS automatically enables "Legacy Mode". Well, apparently that was the problem. As disabling it, everything all of a sudden "works". I didn't think much of it at the time, but after exhausting all other ideas I had, I went over the BIOS with a fine-tooth comb, and took my chances with that setting. It worked, and I'm now on my way to a (FINALLY!) complete system.