You are making some progress with setup of wireless network. Apparently there is reliability problem with the wireless connection. Perhaps you want to refer to following Apple support documents and start tweaking parameters in the router to improve the reliability. I used to have third-party routers such as Belkin and Netgear but switched to Apple Airport routers (Time Capsule then Airport Extreme for now) because of incompatibility issues whenever Apple upgrade Mac OS X. But since Apple develops Airplay protocol as alternative to Bluetooth for wireless audio/video streaming (with much better bandwidth) and must be deployed in any wireless network, there is no reason third-party routers not working for Airplay protocol.
About AirPlay Mirroring in OS X Mountain Lion:
http://support.apple.com/kb/HT5404
iOS: Recommended settings for Wi-Fi routers and access points:
http://support.apple.com/kb/HT4199
Troubleshooting AirPlay and AirPlay Mirroring:
http://support.apple.com/kb/TS4215
Good luck in setting up 802.11 wirelesss network.