I know this answer is likely too late to help but it may help users in the future. In the situation where you are using a range extender there are a few things you need to confirm to make sure it is going to function properly and also some best practices to make sure your network is running smoothly.
1. Is the range extender using the router for DHCP? I have found in my experience is that the router should "reserve" a MAC Address and IP for the the Range Extender. Most times it is 192.168.1.2. I would first set up the router to have the range extender have a set IP Address of 192.168.***.2 and add the MAC Address to the list so that the Range Extender has a Static IP Address in the private network.
2. Initially set the range extender to have an SSID(Wireless Network Name) different than the Router's(Airport) SSID. This will allow you to determine if the Extender is functioning normally without it interfering with your existing network.
3. At this point it is important to set the Range Extender DHCP. What this will do is let the range extender request IP Addresses, DNS(Domain Name Servers), and UPNP Requests. This is where I feel that a lot of people are having issues and assume that the router will still resolve DHCP. If the SSID and Wireless Channel are the same your network will work intermittently with internet access until it is configured correctly.
4. For Devices that will stay on your network such as NAS(Network Storage), Smart TVs, Apple TVs I would recommend setting Static Private IP Addresses for each of them. Such as 192.168.1.3(NAS), 192.168.1.4(Apple TV), 192.168.5(Smart TV). Also if you enter the MAC Address for each device when you set this up it would be recommended.
5. If devices are having issues connecting to internet or certain web services are having issues. It may be necessary to set the Extender(Which is now handling DHCP with IP, DNS, and UPNP passed through from the router) as the Gateway for which device may be having issues. For instance if the Apple TV is having issues communicating on the network(Air Play, Sharing, Streaming) set or configure the network settings manually on the Apple TV. The IP Address would be the one your reserved on your router. The Subnet Mask would likely be 255.255.255.0 by default unless you have changed the amount of potential "Hosts" on your network. The Gateway would be the IP Address of the Extender(Which is paired with your router). DNS servers should be handled automatically. In some situations with certain devices you may need to use a DNS Service for Streaming, but this generally isn't needed.
A Visual Representation would be
ISP>Modem if needed>Router(DHCP Enabled and MAC Addresses Reserved)>Localized Devices<Range Extender(DHCP Enabled and Static IP Address 192.168.***.***)<Apple TV(Manually Configured IP Settings and Range Extender set as Gateway)<Other Devices Localized to Range Extender but not in range of Router(AirPort).
In this representation you should be able to have the router and range extender broadcast Identical SSID and Channels(Radio Frequency). This also allows for advanced settings to be put in place such as LAN networks or Subnets.
An issue that may come up is communicating with the range extender or router while setting up the network. A simple workaround that will allow you to configure either if the network doesn't let you connect on a MacBook is to go to System Preferences>Network>Wifi or Ethernet>Advanced>TCP/IP in configure IPv4>Set Using DHCP with Manual Address> Then IP Address that corresponds with your current network 192.168.***.*** but is not the same as the gateway, router, or reserved devices. This will allow you to connect and configure devices even if router or extender is not assigning IP addresses or operating properly.
Disclaimer* This advice is only for informational purposes with no warranties expressed or implied.
