Thank you for providing all the details regarding the playback issue you are experiencing with your Apple TV. I found an article with some steps you can take to troubleshoot an issue like this.
I recommend reviewing the section titled "If you are unable to access or play content while connected to a Wi-Fi network" in the following article:
Thank you for using Apple Support Communities.
I ran through the networking recommendations, but didn't find that altering anything affected whether I could or couldn't play certain songs. I know the signal strength's not dropping because my Time Capsule is within reach when I'm at my laptop (as my apartment is temporarily set up), and because it'll immediately play another song if I try.
Thanks for your help! I wish that'd been the problem.
Thank you for using the Apple Support Community. Due to the nature of your issue or question you may find more information by using another one of Apple's support resources - https://expresslane.apple.com/GetproductgroupList.action
All the best,
im by no means an expert in any of these issues - i only got my mac tv yesterday and my air a few weeks ago.
It looks like both the mac and itunes have the ability to air play individually but not at the same time i.e. when your itunes is connected to air then you can play music but cant share your screen / when your mac is connected you can share your screen but cant play itunes.
theres an easy way around this - use a third party media player - this way you can share screen by connecting your mac to air and music froma third party player such as VLC will play through your TV and connected speakers.
its surprising the guys at apple couldnt help with this / couldnt identify the issue - ive had tremendous support over the past few weeks getting all my devices working together (reinforcing my point earlier that im no expert, but found a solution to this quite easily).
hope this helps, if ive missed the point of the post entirely then i apologise!