Great feedback -- and questions. So let me see if I can answer them, after working on the problem last evening. As I did mention, the drive shows up in the BIOS. The information about it is entirely correct. I then booted to Safe Mode, as suggested. Here's the interesting part: the drive still does not show up in Windows Explorer. HOWEVER... I can now run Disk Management (which, as mentioned, would just "hang" without ever even opening the Disk Management window before); and the drive shows in Disk Management! I can see all of the partitions (it's a 640GB drive, partitioned into 4 or 5 partitions. And Disk Management shows the drive status as Online. But I just can't see it in any applications.
So I then booted into the Dell Diagnostic Utility (which installs on all Dell systems in a separate partition), and ran a superficial system-wide "quick test". Everything seemed to come back OK. Then I ran a lengthy, in-detail, deep-down test of my hard drives. (I didn't want to just test the one in question; I wanted to see the complete test run on both HDDs.) The test ran to completion on the first HDD; and it started on the second drive -- the one with the problem.
(Note -- it did see the second drive, and even began running tests on it.) The first and second test -- which were just what Dell calls "Quick Tests" -- ran through just fine. The next test was the in-depth Read Test. It went thorugh the first 11% of the test with no errors. Then suddenly I got an error at the 11% mark. It was a "Timeout waiting for IRQ" error; it gave me the block number and asked if I wanted to continue the test. I clicked on Yes, and it went away for about 10 minutes, apparently running the test. (The disk light was on solidly.)
But after 10 minutes, it came back with another error -- the same thing: "Timeout waiting for IRQ". And the block number was just +1 block more than the last one! So once again I clicked on Yes, and it went away for another 10 minutes or so, with the disk light on. Then the same thing: "Timeout waiting for IRQ" -- and the block number was now +1 block more than the last error! It's still running as I type this.
Questions: What is a "Timeout waiting for IRQ" error? What does it imply? Why was it taking 10 minutes to get from one block to the next, if it was getting those errors? Why would it get 11% of the way through the disk (more than 7GB) before this error showed up? Why would this error cause the disk to "disappear" from things like Windows Explorer, but still show up in Disk Management (but ONLY in Safe Mode)? Why won't the disk correct this error, as it does with bad blocks/sectors, etc.? What can be done about the problem? What other diagnostics can I try, to get more information? Is there a way to run CHKDSK or SFC or some other diagnostic? (Clearly not from within Windows.)
Thanks again for the great feedback, and I hope someone can provide answers to the above questions. (BTW, I do have the model number, and other information about the drive. But I suspect that something as simple as a driver update isn't going to affect this.)