I use a VPN service that is connected to a server in another country, however this VPN service does not offer control over my DNS requests to block some sites so I preferred to use another DNS resolver that has this function.
My question is: When I access my VPN’s website, it accuses that a “DNS leak” is occurring. Can the DNS provider know my real IP address, or does it only know my VPN’s IP?
Can the router be configured to not resolve dns
Yes, you can configure non-router DNS in your DHCP server or you can manually set DNS on individual hosts. For a VPN you want to make sure the VPN connection has DNS manually configured.