The web site you are accessing doesn't get to know which DNS server you are using.
yes, and no. Some sites use DNS to route traffic to different servers for different regions. For instance you could have you site hosted on 2 different data-centers. 1 in EU and 1 in US. And then make US-centric DNS route traffic for your site to the US-datacenter and he EU-centric-dns to EU-data-center.