What are the compression/resolution/bandwidth limitations of Airplay mirroring? In a perfect world Airplay mirroring would display identical results compared to an HDMI cable. In other words, the frames per second, resolution, and image quality would be unaffected by the use of Airplay compared to using traditional means. 

What are the current limitations of Airplay?

Can I stream a 1080p movie through Airplay onto my television (through Apple TV)? 
Can I play a first person shooter through Airplay without lag?

Assume a powerful 2013 iMac. 

I would appreciate any information related to frames per second, resolution, image quality, bandwidth, lag/delay, and what factors are associated with these limitations.
The Apple Developer documentation for AirPlay, specifically the HTTP Live Streaming section, has pretty detailed information about it. Depending on several factors, I summarize the following:

Resolution max. recommended high profile encoder settings, for a 16:9 aspect ratio over wifi:


Dimensions: 1920 x 1080
Total bitrate: 40000 kbps
Video bitrate: 39000 kbps
Keyframes: 90


Framerate A frame rate of 10 fps is recommended for video streams under 200 kbps. For video streams under 300 kbps, a frame rate of 12 to 15 fps is recommended. For all other streams, a frame rate of 29.97 is recommended.

There is tons of more info, but in short, the specs say: "Yes!" you can stream a 1080p movie through Airplay. And there are several examples of "Airplay-enabled-games", like first person shooters, that work just great. 

However, the crucial factor will be your wifi network as this will determine your lag/delay and -probably- most limiting factors. So there will be a significant delay on any 802.11N based wifi network. And therefor, there will be lag with latency-sensitive audio/video interactive purposes, like games. For movies, it should be no problem though.

Summarizing: The limitations of wifi transfer speeds isn't near to your perfect world when connected through a cable.