Data rescue implies recovery of data from corrupted drives and drive
imaging or ddrescue will not do that. All these programs are concerned
with are making a copy of the existing drive contents. The difference
being that ddrescue is a byte for byte copy that can be plugged into
the PC and run just the same as the original drive, whilst imaging is
a method of creating a compressed snapshot that can be restored back
to the original drive ( or a new one ). Most imaging programs now have
the ability to mount the image so that the contents can be viewed and
files copied back to the system without the need to do a full restore.
Data Recovery is a whole different ball game

As for the Apricorn Drivewire adapter, this has its own power supply
so the USB side is only used for data. Connecting to a powered hub
shouldn't make any difference as power draw on the USB side will be
low. It's the power draw of the HD motor through the adapter that's
possibly where these larger drives are falling over but not definite
that that's the cause. Later drives of 500Gb and upwards may well have
4Kb sectors ( instead of the 512 byte sectors that HD's have had for a
long time ) but I can't see that being a cause of them dismounting.

A curious problem that might just mean that your adapter is not quite
up to the job with these larger drives, or wear in the connectors is
beginning to make itself felt. Also does the power unit get hotter
than it used to do ?