For the second question nic.* is conventionally the domain utilized by the TLD operator. So nic.brand may contain info and functions related to the .brand TLD and then www.brand and about.brand be other sites. E.g. nic.google redirects to registry.google, Google's domain registrar whose engineers also operate .google, whereas about.google can be considered the actual brand home site.
> E.g. nic.google redirects to registry.google, Google's domain registrar whose engineers also operate .google
Negative. Google's TLDs are operated by Google Registry, as shown on that site, which is an entirely separate team from Google Domains (the registrar).
For basic information, check out https://www.sri.com/hoi/domain-names-the-network-information... and https://en.wikipedia.org/wiki/InterNIC