I've blocked all three in my hosts file, and will watch to see whether CC cries "foul" in one way or another. My browser is Chrome, not FF. FF evidently ignores the hosts file, so maybe, by extension, TB does, too. I'm not sure how that works.
My hosts now:
# Copyright (c) 1993-2009 Microsoft Corp.
0.0.0.0 ncc.avast.com
0.0.0.0 ncc.avast.com.edgesuite.net
0.0.0.0 ipm-provider.ff.avast.com
0.0.0.0 shepherd.ff.avast.com
0.0.0.0 ip-info.ff.avast.com
0.0.0.0 analytics.ff.avast.com
0.0.0.0 a23-220-246-23.deploy.static.akamaitechnologies.com
0.0.0.0 a23-220-246-59.deploy.static.akamaitechnologies.com
0.0.0.0 28.176.160.34.bc.googleusercontent.com
All because of CCleaner's behavior. Maybe superfluous, I dunno.