No is the correct answer to the compatibility question. As Tech mentions from a version of avast you shouldn't get asked this question 'if this is a first installation of ZA with avast.' If however you have previously been asked this question, you will continue to get this question, but it shouldn't be an issue when you answer NO.
ZA should have asked you permission after the avast program update and after you rebooted. If it didn't it hasn't recognised the changes in the web shield (and also Internet Mail) provider.
The usual culprit for not being able to connect is firewall blocking.
ZA on occasion has been known to suffer amnesia when it comes to program control, just enter the program control and delete the entry for the avast processes. Now connect to the internet, you should be asked permission for the ashWebSv.exe Web Shield. Now try to check for your email, you should be asked permission for the ashMaiSv.exe Internet Mail. Now do a manual update and you should be asked permission for the avast.setup avast update process.
avast shouldn't touch your network connections or associated files, so I'm at a loss as to why this might be happening.