Hard drive crashed and can't restore backups

Question

I have an older model Macbook (circa 2006).  The original 120GB hard drive crashed, yes I got the dreaded white screen of death, and I replaced with a new 500GB hard drive.  I had backups of all of my data stored to my Time Capsule, but am unable to restore my laptop because the backups were on OS 10.7 and I receive an error message that my OS must be updated.
The install disks that I was able to reboot with after installing the new hard drive are 10.5.8. I found a Snow Leopard disk (10.6.6) but when trying to run an install received a message that it cannot be installed.
How am I supposed to restore my backups and get to the corre?
This is my current system:
Model Name:          MacBook
  Model Identifier:          MacBook3,1
  Processor Name:          Intel Core 2 Duo
  Processor Speed:          2.2 GHz
  Number Of Processors:          1
  Total Number Of Cores:          2
  L2 Cache:          4 MB
  Memory:          2 GB
  Bus Speed:          800 MHz
  Boot ROM Version:          MB31.008E.B02

Answer 1

If the drive fails to appear in Disk Utility then it's most likely died. You can try this:
Repairing the Hard Drive and Permissions
Boot from your OS X Installer disc. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Installer menu (Utilities menu for Tiger, Leopard or Snow Leopard.) After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer. Now restart normally.
If DU reports errors it cannot fix, then you will need Disk Warrior and/or Tech Tool Pro to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.
The flashing question mark means system files are damaged assuming the drive is OK. In this case just reinstall Snow Leopard. All your user accounts and data will be preserved.