The ashServ.exe shouldn't need internet permission, so I don't know why ZA has that in program control.
That is an ancient version of ashwebsv.exe and you found that on your current avast installation ?
Is your current avast installed in the same location C:\Program Files\Alwil Software\Avast4\ ?
If these are correct you have a screwed installation, sorry for getting technical. You need to do a clean reinstall.
Download the latest version of avast
http://www.avast.com/eng/download-avast-home.html and save it to your HDD, somewhere you can find it again. Use that when you reinstall.
Download the avast! Uninstall Utility,
find it here and save it to your HDD.
Now uninstall (using windows add remove programs), reboot, run the avast! Uninstall Utility, reboot, install the latest version, reboot.