Routers, very often, have a service called "DNS relay" activated by default. This is a sort of automagically DNS discover. If you want to use this service you must indicate as default gateway of your LAN, the IP address of the router, commonly 192.168.1.1
If you want to set up a DNS of your choice, you must disable such service on the router or if possible (and usuallly it is) configure DNSs on the router.
If you choose not to use "DNS relay" you have to set DNS while running netconfig or manually on /etc/resolv.conf.
The problem comes when using DHCP configured network, as dhcpcd (the client) temporarely overwrite your /etc/resolv.conf with DNS stated on DHCP server.
This can be overridden by adding -R option when starting dhcpcd or permanently by adding a line such as:
# Config information for eth0:
to your /etc/rc.d/rc.inet1.conf
If you want your connection started when you plug in the cable there is a demon available called ifplugd
but it requires some modifications on your startup scripts.
It seems to me you're not prone to such kind of thinks...so is up to you.
Hope this helps