Can I use an iPhone to remote desktop to a PC, show the desktop on a TV with airplay, and use the iphone as a touchpad for the remote desktop? I've seen descriptions of running an RDP client on the iPhone.  I've also seen descriptions of using the iPhone as a touchpad/mouse replacement for a PC.  But I haven't seen a description of a combined application that shows the remote desktop screen via AirPlay (to an Apple TV) on a TV while enabling the iPhone to act as a touchpad for that remote desktop.  It would make the iPhone/AppleTV/TV combo a complete remote desktop solution without additional hardware (other than a network).
This works out of the box with AirPlay mirroring and any RDC client such as Screens for iOS. 

Just establish a remote connection then double tap your home button and swipe the multitasking bar to the right. Enable AirPlay mirroring and you'll have what you described.