Chromecasting?

I have the Fire HD 8 tablet, 7th generation, and I have HBO Max and Prime Video apps installed, and I have a Chromecast attached to my television. I can cast stuff from my iPhone perfectly well.

Does the Amazon Prime Video app not allow casting to any other device?

Is there a way to cast HBOMAX from my FireHD to the television? When I click the symbol in the HBOMAX icon, there aren't any devices listed to cast to.

I don't seem to be able to install the Google Home app in order to get my Fire to recognize the Chromecast.

Any help?