Knoppix can't solve all bad hard drive problems. The drive might be so dead that it isn't spinning for example, or the heads might not be moving. On the other hand you might have the very common problem of Windows destroying it's partition. In some of these cases Knoppix can still see files, in others it can't. If the drive is running well enough that Knoppix can see the disk then the gpart command might help recover the partition table. An alternative might be to use a forensic recovery CD like Helix and see if it can read any of the files off the disk (the safest thing to do would be to try Helix first, before letting gpart change anything).

You have come across one of the very bad things about the Dell and similar service warranties. Yes, they will replace the disk, but they take your old one, and you're never really sure where that information ends up. If you have banking , email or any other important information out there (and it's likely that you do, even if it's only an e-mail confirmation of a purchase with you credit card number of an account number with the seller who retains the credit card number), you should really be concerned about what happens to the drive. If you can get it working well enough to write to it I would suggest at the very least running some application designed to completely wipe data from the drive before turning it over (reformatting is not enough), if not hope whoever gets his hands on the bad drive doesn't manage to recover it. There are way too many stories of disks turned in to depots that were supposedly going to be destroyed that resurfaced and contained private information.