I’m not sure exactly what you mean by “network is down” or “internet is down”. If you mean that occasionally you are unable to browse the internet and your internet service/modem/etc appear to be working, then yeah you might have a problem with your DNS service and might want to analyse your 2003 server event logs. Problems with both internal and external lookup suggest a DNS problem to me.

It doesn’t really explain why disabling shields makes a difference though or why the other server is still reachable. I think you need to figure out specifically which shield works around the problem (eg firewall) and use that as a point of reference. Since 2003 server ONLY uses SMB1, it might go towards explaining why the 2012R2 server can still be reached if for some reason the firewall is interfering.

I’m afraid I am not sure I can help much more, your site configuration might need some attention before trying to fix avast :-\