How to keep up-to-date between RN and Native Android Screen about current stack?

You should make two-way communication between platforms. I assume you know how to create and register your native modules in Android.

React-Native -> Android communication.

Native code:

@ReactMethod
public void setCurrentScreen(String screen) {
}

In JS code, you can attach navigation listener to your Router via setting onStateChange prop and calling YourNativeModule.setCurrentScreen() method.

Android -> React-Native communication:

Just send an event from FullScreenActivity to JS with activity state information and subscribe to events in JS.

    override fun onStart() {
        super.onStart()
        sendActivityStartedEvent()
    }

    override fun onStop() {
        sendActivityStoppedEvent()
        super.onStop()
    }

    private fun sendActivityStartedEvent() {
        val params = Arguments.createMap()
        params.putBoolean("active", true)
        sendEvent("activityState", params)
    }

    private fun sendActivityStoppedEvent() {
        val params = Arguments.createMap()
        params.putBoolean("active", false)
        sendEvent("activityState", params)
    }

    private fun sendEvent(eventName: String, params: WritableMap?) {
        reactNativeHost.reactInstanceManager.currentReactContext
                ?.getJSModule(DeviceEventManagerModule.RCTDeviceEventEmitter::class.java)
                ?.emit(eventName, params)
    }

In order to access reactNativeHost your FullScreenActivity must be instance of ReactActivity

JS code:

this.subscription = DeviceEventEmitter.addListener('activityState', event => {
                     console.log(event.active)
                    });

You need to create a native module with two-way binding with:

• A function that RN calls on android, invoked every time the user change a react native screen

• When you receive a video notification, you check the current screen and decide if the FullScreenActivity should be shown or not

• An event emitted by the native module every time the FullScreenActivity shows/hides, and react-native can subscribe to it

when app is background or killed, FullScreenActivity should trigger.

I don't understand the question, You can't present/hide/do stuff when your app is killed

If I understand correctly, you can use Actions.

import {Actions} from 'react-native-router-flux';

Actions.currentScene;

Actions.currentScene gives you the active scene.