Does Apple TV software version 7.9 support airplay?

Apple TV (3rd generation) with system software version 6.1 or later supports AirPlay as used today.

However, content providers can choose to not make any provisions for AirPlay compatibility, or have requirements for later models for AirPlay, or even block AirPlay for their streams. It varies. See their respective sites for details.

AirPlay system requirements - Apple Support

Use AirPlay to stream video or mirror the screen of your iPhone or iPad - Apple Support

If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support