Can a short-circuit damage a hard disk drive?

Yes, it's definitely possible. For example, the screw could short the +5V line to the +12V line and fry the hard drive's onboard controller.

Yes it can.

A hard drive can fail indirectly: often hard drives which are used for long time tend to fail at next reboot/startup (caused by short circuit or normal power switch).

Other option: a surge from the wall socket or caused by induction after a short-circuit reached the drive electronic or controller (see below).

Note that there SHOULD be all kinds of fuses, diodes, capacitors and shielding that a electrical problem does not spread but cheap hardware is commonplace.

Just to mention it, there is also the problem of (logically) corrupt sectors if power fails while writing. This also should be covered by the drive (rotates long enough to finish write) but might not. Such corruptions might result in transient read errors but could need a disk repair or special tools to refresh the sector.

Typically hard to tell the reason, but a screw connecting the mainboards power lines with the grounded chasis (or with each other) does sound like a probable cause for a short-circuit which induced damage to the disk. (And again, it should not :)

One could say: you have been screwed. :)

However I want to add that your symptoms can also mean that the drive died first and short-circuited the computer or that the PSU was overloaded with the startup current demand from the drives.

Yes!

I've lost two HDDs due to a short circuit. It was quite spectacular. An integrated circuit at the bottom of one exploded, resulting in a loud bang, a little fireball and a nice crater: