@GSGregg: Thanks for your reply and the extra info.

GSGregg wrote:
Thanks, Vulcan; sorry it's so long.
No worries - better to have more info than too little.

The new info was certainly helpful to better understand what was happening.
I should point out that my approach would be different to that from
N.C. (Janos). Therefore in order to avoid wasting my time explaining too many details, if you decide to follow his approach instead, I'll just make some comments explaining a few thoughts. To give full disclosure, I'm not a DR pro - I'm an electronics engineer, with many years experience working in a different part of the disk storage industry. So while I am happy with the risk-analysis of what I would do in your situation, I'm not the person who would be doing it, and in any case, it's your data not mine.

Therefore it's got to be your decision what you do.
GSGregg wrote:
What actually happened was, I came back from a pause in a game to find a box, "McAfee agent.dll (or something like that) ....needs to close."
I see that
poehere has suggested that there might be a virus. IMHO the error message doesn't definitely prove or disprove that possibility. In my experience, since virus scanners are doing reads from the disk, they sometimes find other problems (unreadable blocks, corrupt filesystems etc.), and not only find viruses. But, as I said,
poehere could indeed be right, so remember this is a possibility.
In any case, the fact that attaching that disk drive via an IDE converter (i.e. not booting from it) still causes the PC to reboot (after chkdsk messages), makes me believe that even
if there is a virus involved, something
else is involved too - probably a corrupt filesystem, perhaps initially triggered by unreadable filesystem metadata.
I'm not surprised that this disk is no longer bootable - the more that chkdsk runs, then if it renames or moves a critical system file (due to
perceived filesystem corruption - although that
could really have been caused by memory corruption), then Windows can easily become unbootable from that disk.
GSGregg wrote:
I'll try to shorten the story; the only way to stop the 'loop' was to shut off the PSU, and when I turned it back 'ON', the cycle resumed even though the front-panel switch hadn't been pushed.
That's normal behaviour with some PSUs - they treat removing the power via the rear panel switch as a power failure, and automatically power-on even without you pressing the front-panel switch.
GSGregg wrote:
Eventually, the computer ran CheckDisk---again, no input from me---and recovered two or three files, then rebooted before I had a chance to read what they were. After a few more 'CheckDisk' cycles, with the list of 'recovered' files getting longer each time, things settled into their final routine; two black BIOS screens, then a "We're sorry for the inconvenience, but Windows failed to start.....etc."
This behaviour
could be caused by unreadable blocks on the disk - and as I mentioned above, the more that chkdsk runs, the greater the chance that something vital will get recovered as a "lost" filename and then Windows won't boot.
GSGregg wrote:
I was told, 'bad HDD', defective memory (I had upgraded to 2 x 1GB a couple of weeks earlier; right now I'm back on the old 2 x 256MB just in case)
Good idea to go back to the original RAM, in case the new RAM did introduce a new problem - due to filesystem caching, memory problems can corrupt a filesystem. You could try booting from a memtest+ CD and see if you can provoke a fault with the new RAM, if you want to.
GSGregg wrote:
and 'bad motherboard' (obviously it's still good).
I guess you're saying this, because a replacement disk (and new Windows installation on that) is running OK? If so, then while I agree that this
suggests the motherboard is OK, if the problem is actually a very intermittent fault, which only occurs (and corrupts filesystems) rarely, then you may not have tested the motherboard enough to make a conclusion that it is OK. The fault might be very subtle and/or intermittent. Again, I'd just suggest you to be aware that you're making an assumption - which might indeed be true, but an assumption nevertheless.
GSGregg wrote:
The harddrive is barely three years old (mfd. Jan. '08)---I thought they'd last a lot longer than that.
They can last both shorter & longer times than that.

One thing you haven't mentioned is the PSU - a marginal or intermittent PSU can cause incorrect disk behaviour, leading to corrupt filesystems (we've seen a likely example of that recently on this board). I don't see lots of clues pointing in that direction here, but again, I suggest to just bear this in mind.
So having replied to your extra info, I should point out that you have to decide whether you'll attempt DIY recovery of this or not. There is a chance (which is impossible to quantify at the moment, given that we don't yet know if there is a hardware fault with the disk), that the disk could deteriorate, or even catastrophically fail, while you're attempting DIY. Do you feel lucky?

If you go ahead with DIY, my suggestion would involve something similar to what
poehere suggested - clone that disk (either to another disk, or to a file), which will perform 2 functions: (a) the cloning process will find if the original disk is fully readable, and (b) you get a copy of the "problem" disk, before chkdsk does any more mangling of the filesystem.
If you find the disk is not fully readable, then that explains the likely original problem - and all the consequences are probably due to chkdsk being unable to make a consistent filesystem on a flaky disk (as well as Windows being unable to cope with such a corrupt filesystem, even when not booting from it).
If the disk
is fully readable, then the cause is less obvious - bad memory, other hardware fault, virus, etc. etc. all remain possibilities.
I would do the cloning using ddrescue on Linux (e.g. bootable LiveCD). You don't need to (and shouldn't) mount the filesystem on the "problem" disk to do the cloning (and so Linux shouldn't behave as badly as Windows does). Cloning results are generally better with the "problem" disk direct-attached (via IDE in your case), and not through a USB converter/bridge. The target for the clone can be USB-attached (if desired), since we don't expect there to be a need for good error handling, to that target disk. However, controlling ddrescue for best results, is not a "one-click" operation - it depends on what errors are actually found.
You could also use the opportunity after booting Linux, & doing the cloning, to grab copies of the .evt files (event log files) e.g. sysevent.evt from either the clone or the original disk. Then use an event log viewer (I no longer use Windows, but I'm sure I found a standalone event log viewer ages ago) to see what they show of the original problem (if the logs aren't corrupt or wrapped due to new entries), as you originally suggested.
As you can imagine, such an plan needs lots of preparation, some extra storage, some Linux skills, and a careful approach - and, as I said, any DIY process is not risk-free. Human error or hardware failure could occur and make the data either harder to recover (even for a pro) or totally unrecoverable (e.g. if you get the direction of the cloning wrong and overwrite the "problem" disk). I don't promise to have enough time to hand-hold through the whole process, but I hope the suggestions & comments above are helpful. Good luck!