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?
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.