I see a few issues from reviewing your logs.
First there are a few 1603 errors posted when referring to your HPZInstall. 1603 errors normally mean that the .msi does not have permissions to run on that machine. We can work around 1603 by running installations as Admin or by confirming that all the resources are available that the installer needs. Once the installer can find everything it needs it is usually able to continue.
Building on that were a few Batch file error 2 messages. These are not as much of a concern but Error 2 on a batch file is along the same lines as a 1603 with .msis. This error is another permissions error message where the batch file cannot complete the desired actions for one reason or another.
The last thing that struck me as odd was when you were running the HP Diagnostic Utility. It seemed like the Diagnostic was assuming you were using a HP Color LaserJet 2605 for some reason. Because of this the Diagnostic was not able to find the resources that it needed and complete any kind of repair. Perhaps this is shrapnel left over from a previous install?
A few suggestions to try:
1. Re-Run the installation file "As Admin". This should elevate the installer high enough to get around the 1603 errors that were posted in the logs.
2. Manually delete your C:\Program Files\HP and C:\Program Files x86\HP folders. There are a few times where the installer is not abel to complete the scanner installation because resources in these folders were blocked or already in use.
3. Download the latest installation package from http://support.hp.com. This will ensure you get the latest software updates for your current OS.