I too suffered from the Airport Extreme dual status error problem reported by the Airport Utility, i.e. Internet Connection and No DNS Servers.
I started with fully working network with 3 Airport Express (since I love the audio out feature), a ZyXEL cable modem, and comcast service. I wanted to add an Airport Extreme my network to serve movie files to InFuse running on the Apple TV 4.
I purchased a fifth generation Extreme (7.6.4 firmware) and, via Airport Utility, the Extreme extended my current network without a hitch. However, for no particular reason, I wanted the Extreme to be the Base Station in my network, rather than just an extension of the Airport Express that, remember, was working just fine as a Base Station accessing comcast internet service.
When I moved the cat-5 cable from the WAN port of the Airport Express to the WAN port of the Extreme, the Extreme simply wouldn’t detect the comcast service. I hard reset the Extreme. Power cycled the Extreme. On and on. Did the same to the cable modem. No matter what combination of power cycling, resetting, restarting, and so forth I tried, and I tried for several hours, I kept getting the same dual status error on the Extreme. Thought I was losing my mind.
At this point, I just assumed the WAN port on the Extreme was malfunctioning, since reconnecting the cat-5 cable back to WAN port on the Airport Express brought back my internet service just fine.
I eventually found my way to this post. I followed Tesserax’s excellent post to the letter. Still no success. Then I came across CoastRedwood’s post, which was one thing I hadn’t tried.
I hard reset the Extreme, connected the cable modem to the WAN port, launched Airport Utility, and set up the Extreme to create a new network. A wifi network was created, but still no internet access. I then clicked on the Extreme icon in Airport Utility and selected Base Station, Restore Default Settings, and let the Airport Utility do its thing.
Eventually, Airport Utility popped up a window with a diagram and a message to power cycle the cable modem, presumably because the Extreme wasn’t connecting to the comcast servers. The last line on the popup window was a recommendation to “Reset” the cable modem, rather than power cycle the cable modem, to see if a reset worked. So I pushed in the reset button on the back of the cable modem and waited, and waited, and waited. After about 10 full minutes, the Airport Utility reported “Done” and, amazingly, the Extreme was talking just fine to comcast internet servers. DNS assigned just fine. Presumably a power cycle of the cable modem in response to the Airport Utility message would have also worked, but I don't know for sure. I then set my Airport Expresses that was originally my "base station" to extend the network of the Extreme, and that went without a hitch.
So, thanks to CoastRedwood for weighing in. This is the first time I’ve ever seen every conceivable combination of hard resets and power cycling fail until the software fix of "Restore Default Settings" seemed to clear the path for a hard reset of the cable modem to work. Recall in my situation, my network worked just fine whenever I wanted to use the Extreme to extend the network, rather than use it as the Base Station. I can only assume that a hard reset of the Extreme does not get the Extreme to the same hardware-software state as is achieved through a Restore Default Settings via the Airport Utility. Crazy.