u can use that website to check for the wu logs those are the same as for wsus since its the same technique just using a middleman server like the avast ADNM
That is correct.
Also, if you're on a domain environment, I highly suggest using WSUS if you can, it makes keeping the computers up to date a lot simpler!
Without a sample of the windows update log from one of the computers, this is sort of hard to troubleshoot.
As avast does not have a firewall yet, I don't see why it would "block" the windows update site, so that's probably not the problem.
I don't see why it would prevent updates though either. That's a new one. Keep us updated on what you find, so we can help more!