Meanwhile a lot of water has flown into the sea, but I ask again. Is it possible now to install Amazon Prime Video native app to Android TV? I mean this one: https://play.google.com/store/apps/details?id=com.amazon.avod.thirdpartyclient
I guess you still cannot cast Amazon Prime Video into chromecast?
The link you provided is the Amazon Video mobile app. If you sideload it, the app will work but will play only 480P video and no remote friendly UI.
You should be able to sideload the APK from the Shield onto other ATV devices (the latest version I'm aware of is 4.1.121). I did this with my MiBox and the only issue I have is that I can't play 4K content on it (other resolutions work just fine). I have a Shield to play 4K content in the living room and the MiBox is now on a 1080P TV in my bedroom.
Native App for Android TV released: http://www.androidpolice.com/2017/12/21/hell-hath-frozen-amazon-releases-prime-video-app-android-tv/
But currently not compatible with any Android TV device...
For that one no, seems to be phone only.
Nope, still no casting with with the screen mirroring, the app just blocks it.
There's an apk that works on Mi Box.
The shield and Sony android TVs seem to be the only android TV platforms that have prime video, unless you have one of those you're out of luck