Hi,
I see two questions here:
1 - My app not checking mail in the background. Please see your device's system settings, under Accounts and Sync. Most devices have a "switch" there that forces off all background syncing. Please check. Alternatively, see my app's settings, under Receiving mail, and change "System settings" to last option, "... even when background data is off".
2 - The unread number not showing on the icon. Please check that you've enabled "External notifiers" in Apex Pro settings (this is under Advanced settings in Apex).
Also, IIRC, Apex Launcher has to be installed first, and my app after it -- so that my app can acquire the permission to send counter updates to Apex (the author of Apex set it up that way, if you think it's a hassle, please tell him).