Sorry but your Norton DOES NOT show that is active and no firewall is showing as active either.
You HJT log shows multiple Norton files which are supposed to run as a service when your computer boots up are missing, therefore, it is not running.
If I were you I would completely Uninstall Norton and then Reinstall.
Now in order to do this you will need to go through Add/Remove and uninstall all instances of Norton AND Symantec that are showing. But since some files are missing you may have to use your Install disk first, reinstall and then your should UNINSTALL it all. After doing this then reinstall, update and enable it.