Pardon my confusion of terminology. I think I would lean more towards ddrescue
than an image. When trying to rescue data, does one method have an advantage
over the other?

That USB to SATA adapter (Apricorn Drivewire) has been tested with several
drives. The problematic drives were all purchased within the past year. They
have been of various makes (Seagate, WD) and sizes (500G and 1TB). The 1TB
drives of more than a yr ago didn't have the problem of dismounting, but the
later ones have. Also, I've tested several older IDE drives, and there's not a
problem with them.
I've disconnected other USB devices and have tested on multiple machines and
OS's (XP and 7), but the behaviour is the same on each.
I've re-tested it on a drive(s) known to be good, and it works fine.

To correct the voltage problem, would a powered USB router help, possibly? Or,
is there a good measuring instrument to somehow detect when (or where) there is
an interruption in power supply? Maybe more accurately, an instrument to detect
when the HD's demands exceed the supply?