Which is the firewall on the remote machine?
And in yours?
I’ve used LogMeIn without problems between Vista and XP… but I never used PCAnywhere.
Sorry, can’t help further…
This program is not a firewall and it does not block connections. However you are (understandably) focused on the fact that you installed avast and something changed.
What you may not be thinking about is the possibility that the Windows operating system has let you down. I too once found that installing a product made it impossible for my systems to connect on my home LAN.
For me it was not avast that appeared to be the cause of the problem but my choice of imaging software. However it turns out that - load enough packages and Windows will run out out of a necessary resource and not manage the problem and not tell you it is a Windows problem.
While your report is a little different from others who have found this solution works (since you have a third party product involved) I hope it assists you too.