I understand that you have some issues with your Slingbox since you can use the SlingPlayer for Android Tabs only when on your local Wi-Fi network.
This issue seems to be related to the Internet Viewing feature on your Slingbox. I recommend you to verify if you are able to get the remote connection to your Slingbox using your computer and from a different network. If you are not able to, you will need to run the Setup Assistant for Internet Viewing.
The following link will provide you with the steps to follow:
Hope this helps!