I'm not sure how you have the drives connected, but say the customer's HD is /dev/sda and yours is /dev/sdb. Browse the filesystem to make sure you know which is which.
When you mount the customer's HD, use the 'readonly' option. The disk will not be touched in this mode.
If you're not sure if this worked, just look for a safe place on the HD and try to save 'blank.txt' or such.
sudo mount /dev/sda /where/you/want/to/mount -o readonly
2) The destination does not have to be formatted. The easiest & safest way IMO to do it is to make an iso file.
The dd command will be:
So the HD will be backed up to an iso file, which you can mount in any filesystem and look over in any operating system.
dd if=/where/you/mounted/HD of=/where/you/want/backupfile.iso
It could take a long time, and dd gives no status messages.