Can an Apple TV 2 be operated without HDMI plugged in?
I know for sure Apple TV works fine with AirPlay to send audio out the Toslink (digital optical) with or without a HDMI cable connected.

The converter you linked to seems to take RCA (electric) digital signal and turns it into Toslink (optical) digital signal so I doubt that's exactly what you want to get to a headphone jack with analog stereo (electric) signal.

Since it's not clear what input your stereo wants, it's unclear if everything will work, but the Apple TV will work as you ask.

I have this setup (but Toslink to my speaker) as well as a second stereo connected via bluetooth and a Belkin receiver that connects to the normal 1/8" headphone jack. That might be cheaper than cables and an adapter if the distance is 30 feet or less.