Q: Apple 4 TV

What model # TV?

As I understand you have the HDMI connected from the ATV to one of the two HDMI inputs on the TV and an optical cable from the TV to the input marked TV on the receiver.

FIrst your TV does pass Dolby 5.1 through the optical output.

On the receiver rotating source select to TV with the default audio setting of digital should output the ATV audio.

I would go into settings on the ATV4 and under audio set the surround settings to Dolby Digital and not Auto since the TV is not a Dolby Digital speaker system.

That should do it and force a Dolby Digital signal through the TV.

I did not notice the 3rd HDMI on the side of the TV.  Sorry for the confusion.

With the optical out from the TV plugged into the TV input on the receiver, the receiver knob for source select on the left front needs to be set to TV.

If you have other inputs such as your cable box connected to HDMi 1 or 2 on the TV, again the receiver must be set to TV to get the audio coming through the optical out no matter which TV source you pick.

Ok, now confirm your TV plays the audio fine through its speakers yes?

Also I assume your receiver plays fine from other inputs yes?

If so, then we eliminate the ATV and cables and focus on the TV optical splitter or the receiver settings or input.

Please post a picture of the receiver display when attempting to use so I can see the status of any setting indicators

From your picture you have your satellite box hooked directly to your receiver and an HDMI cable hooked from the monitor output back to the TV.

I would suggest the following:

Eliminate the optical cable and instead hook the ATV directly to the receiver with HDMI to one of the 4 HDMI inputs and then set the input knob to that named input.

Yea!

The whole time I assumed you had all your inputs hooked to the TV and were just trying to get audio from the TV over to the receiver via the optical output.

A picture is worth a thousand posts...

First, need to discuss surround sound.  When your receiver is displaying 5 channel stereo it is playing the same sound from all 5 speakers.  This is a mode of the receiver intended for plain stereo signals.  When your receiver displays Dolby D it will play the specially encoded audio from the ATV which actually plays different sounds from the different speakers to give a 3d effect.  This second mode is what is known as surround sound commonly.

If you want to force your receiver to play 5 channel stereo instead it can be done.  Please post a picture of the back of your receiver so I can see all your connections to give specific directions.

On the Airplay issue, please post the following information to begin the discussion:

What specifically are your trying to do with Airplay?

What specific  device are you trying to Airplay from?

What is the internet connection that your ATV and the Airplay device are connected to?