Security certificate errors can be frustrating to deal with, I have had to troubleshoot them on my own computer a few times. I am not sure which suggestions you tried from the wrenchmen in the other posts. Here is an excerpt with common fixes for certificate problems from the Wrenchmen's Toolbox sticky just in case you didn't see it:
Originally Posted by Aregionius
I get a certificate error:
1. Your computer's clock my not be listing the correct date, Make sure you sync your computer
to the actual date
2. Turn off Certificates in IE, IE Options > Advanced > Restore Advanced Settings, IE Options > Advanced > Uncheck "Check for server certificate authentication"
3. Make sure your Windows Settings
are properly configured.
4. Update your Internet Explorer, or if you have a more updated version of Internet Explorer you may have to uncheck these settings instead: "Check for server certificate revocation," "Check for publisher's certificate revocation," "Warn about certificate address mismatch."
I see in your dxdiag log that your date appears to be correct on your computer, so that is probably not causing your issue. However, it is worth a try.
I would recommend that you visit the League of Legends support page linked in step 3 and double-check all your Internet Explorer settings. Here is the link again:
Hope some of that helps you get you back online.