AVAudioSessionCategoryPlayAndRecord with Airplay
What AirPlay device are you trying to use? Does it have a microphone?
If not, iOS won't present it as an option when using the PlayAndRecord category, because that device can't play and record. It would show up when using the Play category though.
Also, if the device you are using is a Bluetooth device, have you set AVAudioSessionCategoryOptionAllowBluetooth to YES?
From the Audio Session Programming Guide › Working with Categories › Choosing Categories and Modes for AirPlay:
Only specific categories and modes support AirPlay. The following categories support both the mirrored and non-mirrored versions of Airplay:
AVAudioSessionCategoryPlayAndRecord category only supports mirrored Airplay.
If you want to change where the audio is going you need to call AudioSessionSetProperty, and pass it the constant specifying where you want the audio to go. These constants are
kAudioSessionOverrideAudioRoute_None, which specifies that you wish the audio to be routed to the receiver (the airplay if that's where it is going), or
kAudioSessionOverrideAudioRoute_Speaker, which specifies that audio should be routed to the speaker at the bottom of the phone.