EA4500 behind NAT Router - would cloud config work?

Question

This is the topology:  
Internet +--(coax)--+ Modem/Wired Router Combo +--(cat5e)--+ EA4500
Would it EVER be possible to configure the EA4500 with CiscoConnectCloud?   How could the cloud reach through to it?  
I have an EA4500 that I have never been able to associate with a cloud account. 
With another wireless router, all I did was turn off DHCP service on the wireless router, and plug the upstream ethernet cable into an ethernet port (i.e. the cat5e goes into the blue, not yellow, port.) and let it rip. 
  I wouldn't deliberately choose cloud config for this, but I am here.  Is it even possible?

Answer 1

Thank you for your careful answer.  These are really good suggestions, and I bet they will help many people.   In fact, at my house I should implement them even though my current best router is a Netgear N600 type.   That sounds bizarre, but it is because this EA4500 isn't mine. 
It belongs to my parents, a thousand miles away.  Last April when I visited them, I set them up with a Roku XS.  They have necessary activities going on their computers, active trading and such.  Because I'm not there, at my recommendation, they rent their networking stuff from Comcast, so Comcast can help them if there's a problem.  They live in one of those houses that originally was one fourth of its current size, so they have different density walls, many different base electrical circuits: not what is best for wireless networking.  Their established wireless setup couldn't begin to deliver to the TV.
I knew it was rather insulting to the EA4500, but I bought it and set it up solely to deliver content to the Roku.  I also figured if their wimpy wireless went out and Comcast made them wait, I would just tell them how to connect to the Cisco, as a backup, if necessary.   They didn't use the Roku a whole lot, and for a while they thought they did something wrong.  So it was a while before I even heard about their issues. It was a very early unit, purchased on April 9.  I may also have jumped to a conclusion - they said it stopped working in the week or two before the Fourth of July. , Dad and I did a lot of sleuthing over the phone, on and off for weeks.  Then  I assumed it was the firmware update that broke their connectivity.  So this week when I had them ship me the router so I could work on it at my house.
Your ideas are great and I am going to set it up here the way you both say and let the Cisco do the heavy lifting.  That will tell me for sure whether the thing is working properly. 
Because for right now, the only way it works as a router is when it is dumb and connected via LAN not WAN.  It's the only way I can log in to configure it, and it's the only way I can connect to the internet "through" it.  I've been going back and forth between the firmware versions, topology, and wireless versus wired.  And resetting.  Gosh, thanks for getting me out of that rat trap!