1. When the app is in the background, the received push notifications are handled on operating system (OS) level - an OS-specific notification is shown.
However, when the app is in the foreground you are responsible for handling the push notification event by using the notificationCallbackIOS, notificationCallbackAndroid, or notificationCallbackWP8callback function on iOS, Android, or Windows Phone respectively as described in the documentation here.
Therefore when you tested and the app was in the foreground, the functions in the push notification sample hybrid app show the full content of the notification object received, whereas when the app was is in the foreground, the notification is received but handled by the OS and only a specific notification message is shown in the notification tray/notification center.
2. Clicking the app icon does not show the push notification - the push notification even described above is not always available as described in the documentation. For instance, when you have received a notification (it is in the notification tray in Android or in the notification center in iOS) and you click on the app icon, the event is not available and therefore the notificationCallbackIOS, notificationCallbackAndroid, or notificationCallbackWP8callback are not executed.
This behaviour is by design of the native platform (Android / iOS) and cannot be altered.
3. An icon badge count is an iOS specific feature that is not available by design on Android.
I hope this has helped.
Telerik by Progress