Simple Question: Can a device recive a push message when the app is not running (native push) with ionic push service?

You're still in Debug mode. Prepare for native pushs (still coming from Ionic.io) and everything will work. You'll need to get some Tokens and Ids by the develop center of Apple and Google (for iOS and Android).
https://developer.apple.com/
https://console.cloud.google.com/home/dashboard