If you can back up the .apk of an app, it will work on any other device that is verified with the store it came from and the same account. Each appstore usually uses their own protection scheme. Since Google Play is not installed on the Fire, even sideloading it won't work. But once you sideload a non-"supported" app to the Fire, if it verifies through the Amazon appstore you are good to go.
So yes -- download an Amazon app with Amazon account X to any Android device that allows you to, and the .apk will work on any other Android, by sideloading it and having it check in with the Amazon appstore.
The issue with the backup you did was, that there was nothing wrong whatsoever
except that Google Play was not installed on the Fire.
tl;dr
yes.
yes.
no -- yes.
You don't have to, you just need to do the original backup on the Amazon appstore-bought version.