Do Apple TVs work only with Apple routers?
The Apple TV is a client device, meaning it connects to the router. However, the current generation Apple TV does not have the capability to extend the network; i.e. act as a router.
Edit: To answer the question you edited in, yes, you can control the Apple TV from any Wi-fi network, as long as it is a 802.11a,b,g, or n network. Wi-fi is Wi-fi, no matter what device is broadcasting it.
Join: yes. Any 802.11 a/b/g/n network should work. Extend: no. The Apple TV can't extend any wireless network.