Imac 6,1 intel based core 2 duo (running Lion) will not boot.

Question

I hope this isn't too much insight - I have never posted (not being very good with computers and not having had a problem before) - but often see frustration when someone posts sans detailed t-shooting information.
My 2006 model Imac (Intel core 2 duo, 2.16GHZ, 4 MB L2 CPU cached/shared, running OS 10.7 Lion) suddenly shuts down during each boot.  I still have my original installations disks, plus the Leopard and Snow Leopard disks on hand).    I do not have Time maching but have a 500GB external HD I had before Time maching came out.  Alas- it was a Sipleshare NAS drive - and I have some items on the Imac not back-up that I need (I know.. hindsight stings).
So - in order, here is what I have tried so far:
Step 1 - Apple hardware test:  installed orig install disk, held down D as Imac booted.  Ran test - all successful. 
Step 2 - manual that came with Imac said to:  press power button, ad then simultaneously press Command, Option, P, & R keys until I hear start-up sound a second time.  Heard the 2nd start-up sound.  It starts to boot, a progress bar starts, and then shuts down.
Step 3:  Tried to set Imac in Target mode (using my 2009 Macbook Pro as host with 800-800 firewire with these instructions (http://support.apple.com/kb/HT1661) ) to get to the data and save.  No dice.  Imac displayed the slowly bouncing target mode icon (which excited me) - but Macbook did not see a firewire connection.
Step 4:  Tried an 800 - 400 firewire connection with target mode (since imac supports both) just to try another cable.  Macbook still does not see firewire connection.
Step 5:  I inserted my original Mac OS X install disc 1 that came with the Imac in 2006 and held down the "C" key.  It booted to the "Mac OS X Installer" screen.  From there I ran Disk Utility - with the following results:
Step 6 - verify disk:
Verifying volume "Macintosh HD"
Verifying HFS plus volume
Checking Extents Overflow file
Checking Catalog file
invalid node structure
The volume Macintosh HD needs to be repaired
Error:  The underlying task reported failure on exit
1 HFS volume checked
     volume needs repair
Step 7:  Tried to repair disk
Verifying and Repair disk "Macintosh HD"
Checkig HFS plus volume
Checking Extents Overflow file
Checking Catalog file
invalid node structure
Volume check failed
Error:  The underlying task reported failure on exit
1HFS volume checked
     1 volume could not be repaired because of an error
Step 8  - within this same "Disk Utility" - if I select (from top menu) "Images" and "verify" - or "convert" -  I can browse and actually see my files - but they are excruciatingly grayed out (tantalizingly close but unreachable).  I would love dearly to be able to access these files but I have as yet not cracked that nut.
Step 9 - Selected "utilities - Startup Disk" - and three options are avail:
      Step 9A:  tried starting from HD (which I knew would not work, but tried just for t-shooting thoroughness)
       Step 9B:  tried starting from "Mac OS X, 10.4.8 install disk 1" (which just loads the same "Mac OS X Installer" screen I was already in).
      Step 9C:  tried starting from the final option "Network startup" (I have an ethernet connection to my router for internet, which I know is online as this Macbook Pro is connected to the same router).  Once I restart in "Network Start-up" mode - a flashing globe appears for a minute or so, followed by the Apple logo.  A progress bar appears below for about 30 seconds, and then I simply see the Apple logo with the gray "wheel" spinning below it - even 90 minutes later with no sign that it is succeeding.
Summary:  I feel like I have exhausted whatever I can feebly attempt, but I'm a neophyte when it comes to computers and all of this has been just me searching, reading, and methodically trying one thing after another.  I'm hoping that the far smarter here in this forum who have had the patience to read all of this might have a recommendation that will at least allow me to salvage the data I can "almost" see when in Disk Utility/Image/Verify browse mode! Thanks so much for the help and insight!  Joe.

Answer 1

Here are the full OS X Mavericks: System Requirements
Learn about the system requirements for OS X Mavericks.
To install Mavericks, you need one of these Macs:
iMac (Mid-2007 or later)
MacBook (13-inch Aluminum, Late 2008), (13-inch, Early 2009 or later)
MacBook Pro (13-inch, Mid-2009 or later),
MacBook Pro (15-inch or 17-inch, Mid/Late 2007 or later)
MacBook Air (Late 2008 or later)
Mac mini (Early 2009 or later)
Mac Pro (Early 2008 or later)
Xserve (Early 2009)
Your Mac also needs:
OS X Mountain Lion, Lion, or Snow Leopard v10.6.8 already installed
2 GB or more of memory
8 GB or more of available space