iptables default settings should not affect this at all, so unless you have done something to iptables, they should be all fine.
To request a IP from the router using DHCP you need to issue the command: sudo dhclient eth0
If you can ping the router successfully, that means that the connection to the router works fine, but the connection further than that don't seem to work then.
If you can ping 184.108.40.206 but can't ping google.com that means that the connection to the internet works fine, but DNS don't, most probably due to misconfiguration, in which case you should try adding to /etc/resolv.conf the three ip's you mentioned above. If that wont work, try using following ips: 220.127.116.11 and 18.104.22.168 (those are googles public DNS ips which are known to work, I am using them right now)
If you can't ping even 22.214.171.124 but can ping your router, that means that your router refuses to route your connection to the internet. This could be caused because you use a static ip, instead of requesting one from the router using dhcp.