Before anything else, during all your troubleshooting steps, use AC power, as oppose to using the battery only. After that, keep reading.
If in PMagic you can't see your HDD, I would like to suggest checking the BIOS (as mentioned in a previous post). The BIOS should be able to, at least, recognize that a HDD is connected.
If the BIOS can't recognize it, change its settings from AHCI to "IDE compatible" or "disabled" (or something similar in that same category) and reboot again into the BIOS so to recheck if it can recognize the HDD.
It is unlikely that PMagic (or any other tool for that matter) could recognize the HDD (so to make your backup) if the BIOS can't recognize its presence.
Independently of the first booting device (meaning, no matter if you select to boot into NET, UFD, CD or your main HDD), the first screens during the booting process ("way" before Windows or any OS starts) should show you information about the motherboard, the BIOS, etc. Among others, one information you should see is your HDD listed. Again, this may depend on "AHCI", "IDE" ("disabled") or some other BIOS setting.
Sometimes those first screens are "hidden" by a "computer's logo". This setting is also changed inside the BIOS. If those screens are "too fast" to see, press the "pause" key on your keyboard. Press "enter" to continue.
My point is, get the BIOS to "somehow" see the HDD (even if it can't boot it), or it might be difficult for any tool to do a backup of your files while using that same computer.
Now, about PMagic, the first icon you see in the PMagic's desktop is a file manager, similar to "Windows Explorer". So to see your HDD and your UFD for backup, you use it almost like Windows Explorer. Be sure to have enough free space in your UFD to copy all your files. I suggest to AVOID copying all the 6000 files in only one pass, just to be sure about the available space and to avoid using “too many” resources (RAM).
Let us know about your improvements, and if you need more specific details about some step, or if you see any errors. There are many different scenarios, so if I write them all I will only confuse you.