What are the compression/resolution/bandwidth limitations of Airplay mirroring?
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.