Must every activity have a layout?

The answer is yes it’s possible. Activities don’t have to have a UI. It’s mentioned in the documentation, e.g.:

An activity is a single, focused thing that the user can do. Almost
all activities interact with the user […]

(see http://developer.android.com/reference/android/app/Activity.html)

Related SO question: https://stackoverflow.com/a/12817384/534471

To e.g. display a Toast from an Activity without layout you would define the activity in your manifest like so:

<activity
    android:name=".MainActivity"
    android:theme="@android:style/Theme.NoDisplay">
    <intent-filter>
        <action android:name="android.intent.action.MAIN" />
        <category android:name="android.intent.category.LAUNCHER" />
    </intent-filter>
</activity>

The code would look like this:

public class MainActivity extends Activity {
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        Toast.makeText(this, "I'm alive", Toast.LENGTH_LONG).show();
        finish();
    }
}