No, they aren't compatible. This took a bit of experimenting. Apple TV does not directly work with Amazon Video. So I tried running an Amazon movie from my iPad using airplay. The audio came through, but not the video. Next, I set up my iPad for mirroring and tried to view the Amazon content. This time it popped up a message saying that this app will not connect using HDMI. So I think the answer is no in every mode of operation.
Don't think so on streaming Amazon - I haven't tried it yet for some way into Amazon - but I did read where others said that it wasn't supported - - - so I guess my best answer is a very helpful - I don't know ~:0)
Yes. When you install, there will be a new option on your devices to share w/ TV. Whatever you can see on your computer, iPhone, or iPad, you can easily view it on a large screen. This is true for: Facebook, YouTube, Amazon, or any video clip you can play on your computer (DVDs, blue ray, etc.) I hope that helps.