I am having the exact same issue mentioned here and with all posts below.
There is 1 exception. I can stream Netflix using the Digital AV adapter cable to HDMI, however any downloaded content from Netflix will not play on the TV. That is when I get the sound from the TV but the video is only on my Apple device. Also there are a lot of Ipad users here and I just wanted to let you all know that the same issue is there for iphone (6S plus).
Amazon video has no issues whatsoever...streaming, or downloaded content. Amazon is the best by far!!!!!
Showtime anytime gives you a screen saying that output to other devices is not supported. This is for streaming or downloads
Cannot speak for HBO Go but HBO Now streams perfectly (however they do not allow you to download content at all)