I'm not sure if with the latest version with self protection, terminating or switching off may not achieve what you hope for. Also the new Anti-Leak protection also throws up numerous pop-ups for DDE, DNS, TCP/IP, etc. all of which could effect the ability of ashWebSv.exe to do its job.
Turning on detailed logs should have absolutely no effect on the ability of the web shield to connect, so I thing it is entirely co-incidental.
What plug-ins do you have enabled ?
Does your ISP require that you use a proxy ?
I have to admit to clutching at straws as I can't see any obvious problem, but if you want to humour me and delete the ashWebSv.exe entry in the application section of outpost, if for no other reason than to confirm it doesn't make a difference.
Otherwise I would suggest you try, an uninstall of avast, reboot, install, reboot.
It would probably be best to download the latest version of avast and save it to your HDD, somewhere you can find it again. Use that when you reinstall.