Didn't get an Android TV yet, but if the app is compatible for android tv OS you shouldn't have a problem to install it: use play store directly or dropbox/ES file explorer to access your APK.
On the casting topic, you should give it a try to BubbleUpnp app, it's brilliant with streaming - definitively worth the money. Instead of mirroring and draining the battery of your device it sends the local/streaming link the TV via BubbleUpnp.
There is plenty of information on the topic online.