How can I create a custom home-screen replacement application for Android?

Writing your own home screen application is possible. It is called the Launcher.

You can get the source code of the default Android Launcher via Git.

The project URL is:

https://android.googlesource.com/platform/packages/apps/Launcher2.git

You can get the source code like this:

git clone https://android.googlesource.com/platform/packages/apps/Launcher2.git

That will create a directory called Launcher2 for you. Now you can get cracking and create your custom launcher.

_{If you need help with using Git then checkout Git's documentation section.}

The specific Intent to make your activity the Home screen is:

<activity....>
<!-- Put this filter inside the activity to make it the Home screen -->
    <intent-filter>
        <action android:name="android.intent.action.MAIN" />
        <category android:name="android.intent.category.HOME" />
        <category android:name="android.intent.category.DEFAULT" />
    </intent-filter>
</activity>