Q: TV keep disappearing from my screen/options?

They do have experts, it is called Apple Care. You can call them if you like. Otherwise, you wait for someone that would know the answer to your question, and then respond to you.

Apple Support Communities is a place to find and share solutions with Apple users around the world. It's not about, "customers doing the work".

There might be something in this support article to help with the Apple TV / AirPlay issue >  Get help using AirPlay and AirPlay Mirroring on your iPhone, iPad, or iPod touch - Apple Support

These are user to user forums. That is it. Plain and simple. As it was explained in the Terms Of Use that you agreed to. If you wish to talk to Apple try here

http://www.apple.com/contact/

As for your question, sorry but I am not a paid Apple Expert here, so good luck.

Cheers

Pete