So it looks like the Kindle Fire supports sideloading, which is great, and supports it even without the use of ADB - a user can download an APK and tap on the "download complete" notification just as on any normal Android device.
However, when I actually try loading my app this way it looks really crappy in the Kindle's app launcher screen - it's kind of low-resolution in the "Apps" screen and it's extremely low-resolution as a recently-opened item on the home screen. It appears to be using the MDPI version of the icon in both places even though HDPI and XHDPI versions of it are available, but even one of those would look fuzzy next to the icons for the built-in gallery app / book covers / etc.
So: does anybody know where in my app (an extra manifest line, maybe?) I can embed a high-resolution app icon / book cover / etc so that the Kindle will use that instead of a crappily-upscaled MDPI icon?
Note: the following info is now outdated.
Apparently they're doing something funky. From the docs:
How do I get a high-resolution icon for my app to display on Kindle Fire?
The icons on Kindle Fire are dynamically pulled from the Amazon Appstore for Android. Side-loaded apps cannot use this feature. Once you've submitted your app and it's live in the Amazon Appstore, your app's appropriate icon will automatically be displayed.