I know that Azure private DNS zone should be linked to the VNET.
Does it apply to the Azure DNS zone as well? Should we integrate the azure dns zone with VNet? If not, how does the request reach the Azure DNS zone for name resolution?
Only Azure Private DNS zones need to be linked to Virtual Networks.
Azure DNS zones are supposed to be used for public name resolution so you would probably own the domain name and point it to Azure DNS resolvers so that the DNS records that you add to the zone can be queried publicly.
I found this post to be helpful.