If you are launching the macOS installer from within macOS (even when booted from your external macOS 10.8 boot drive), then you need to use the current date & time. The same should apply if you created a bootable macOS 10.11 USB installer and are booting from the macOS USB installer.
The error in the second picture is usually resolved by setting the date & time appropriately. The date & time should either be the current date & time or possibly an earlier date & time from 2017 (or perhaps an earlier year). As far as I know using the installer you just downloaded should in theory have the computer using the current date & time while booting from recovery mode would likely require an earlier date & time from 2017 (perhaps earlier).
As for the installation log, you should just start at the end of the log and scroll backwards to look for the first error which triggered the installer to abort. This can be a bit tricky to find since the installer will mention a lot of "failures", but you would need to figure out whether it was a critical failure (usually the critical failure will show log entries beginning the abort process). Sometimes even when you discover the error which aborted the install, it may not make any sense. The only way to post logs is by using the "Additional Text" icon, but even that probably won't allow the full log to be included and would be unnecessary anyway since usually the last part is relevant as that would be the section where the failure would be mentioned. Also, before posting the installation log you need to follow the suggestion Apple already provided to you. Since the first picture mentions the installer may be corrupted, this would be a good place to start as the error message tells you what to do (don't ignore an error message which provides suggestions/instructions). If it happens again, then it likely indicates a hardware issue of some sort (memory or a failing drive are most likely).
To eliminate any third party software from interfering, boot into Safe Mode to try to run the macOS 10.11 installer. You can boot into Safe Mode even when booting from an external drive, but you must first Option Boot the Mac....select the external boot drive, then immediately (within a couple of seconds hold down the Shift key and continue holding it until you log in).
If you are attempting to run the macOS 10.11 installer while booted to your macOS 10.5 drive, then it will most likely fail because the macOS 10.11 installer requires macOS 10.6+ as mentioned in this Apple article:
OS X El Capitan - Technical Specifications
If you instead have a macOS 10.8 boot drive, then you should be able to run the macOS 10.11 installer. Please make sure to be careful in listing the version of macOS you are using since I'm not sure if you have both a 10.5.8 and 10.8 boot drive here.
Creating and using a bootable macOS 10.11 USB installer should also be considered since it would eliminate any possible software issues with any of your macOS boot drives.