Well even though I dont get OPs message fully, the same happens to me.
When Avast runs, Chrome can't download anything (failed - network error), Battle.net can't fetch files, and I can't download any zip file that will arrive intact(checksum error, thus cant open).
Alot of game patchers don't work anymore, Guild Wars 2 for example, Star Citizen and so on.
It only works if I disable all modules, does not work if I only disable web shield and FW.
Happens since the new version: 12.3.3154.23