Check if value in cloud DB changed while app is NOT running and create push notification?

Which ones? App Inventor apps do not run in the background without using an extension.

You can try Open Source • Background Tasks: Itoo

Then you want a push notification :cry: ,, another extension is required and it may not work as you expect.

Maybe but possibly not.. You could use a StayAwake routine instead of running in the background and just check if a value in cloud DB has changed use if statements and the provided CloudDB tools.