Maybe I'm missing something 
If someone were to examine my computer,
would it look as though I accessed these sites daily?
Nope - your computer doesn't access the site, there's no communication with the site, no data are downloaded from there.
Just your DNS server is asked to convert the particular domain to the corresponding IP address - but that IP address is not contacted.
Maybe I don't quite understand it, but why does avast! have to resolve unrelated address through DNS if user doesn't actually try to access those specific webpages? I understand that if you visit one of the mentioned addresses, avast! checks if everything is fine, but why does it have to if you don't visit them?
This is not connected with the ordinary Web Shield scanning. The Home Network Security feature tries to find vulnerabilities on your local network (say a router with the ROM-0 vulnerability, a router with a weak default password - accessible from the Internet etc.). It also tries to detect other problems like compromised DNS (be it a router problem, hijacked hosts file or something else) - part of which is checking (= resolving) a number of popular domains and somehow evaluating the result; if HNS concludes that the DNS returns suspicious results, it will notify you about the problem. [Like it or not, but those are quite popular domains - but it's certainly not a complete list of what's checked.]
So this is not about the scanning of a particular network connection, this is about evaluating the general state of your DNS. Whether Web Shield would detect the malicious content after your DNS redirects you to a bad site... well, maybe/hopefully. This is just another, different protection layer.
