Hi all,
Maybe if I relate my experience this will clarify your issues.
1. I use a Billion router for my Wireless anyone who has used a Billion will realise that playing with the the security and some of its features can cause lots of issues with connectivity. For example packet filtering can really clobber your access even for someone who knows what they are doing. Secondly Intrusion detection has caused me numbers of intermittent disconnects and blacklists of https sites.
2. I have recently started using FaceTime and for older members in the family I wanted to put it up on my Apple TV. So after getting my Apple ID correct for my Mac, iPad and iPhone all with the correct software versions my iPad would not work but Facetime would. Recognise there is a connection here, the Apple ID is correct.
3. Well if you do not see the Airplay icon you can guarantee that the device cannot connect to the AppleTV. So if you put the Apple TV to sleep does it connect to your router when it wakes? Well if you cannot see the Airplay icon you can bet your AppleTV has gone offline. So for a Billion, you have to forget the network and connect again. See the icon you are working again. So the router and most importantly its reputation for buffering, connectivity speed etc make all the difference. Lets face it an Apple Airport is a Lucent buy in.
4. Next if you can say connect your Mac Computer and iPhone but not your iPad chack your Apple ID for iCloud if it is not the same on all devices it wont work, obvious you say. Well if you have changed it and still cannot see the icon try turning off the iPad. The icon will probably return.
5. In my case I have a screen saver which is using my iPhoto library for random pictures, do you get Bad connection errors, look at your router.
Hope this helps