Quoted from Frogman:The Linksys EA600 needs to be put into bridge mode. If the router has that specific setting, it should work. If that setting isn't shown, you should turn of DHCP. The EA600 is going to try and assign IP addresses, which will conflict with the other router. There are other reasons why your existing hardware isn't optimal.
This is 100% correct. Currently you have two DHCP servers on your network which will cause problems. Either the EA6000 needs to go or be put in bridge mode.
You setup will work but the AppleTV is behind a double NAT right now and is going to have issues communicating with other devices on your network. Chances are they are on the same addressable range so inevitably you’re going to run into an IP address conflict.
Easiest solution is to pick up a cheap switch and swap out the EA6000. Something like this would work just fine.
Single cable from modem to Internet port on router, single cable from router to switch and all wired devices plugged into the switch.