I have the Due working with ADK using both the example sketch and an application I wrote originally for the Mega ADK. The example sketch was just to compile and upload, then the phone would pop up a window on connection, giving me the option to open the URL given by the accessory code.
I used a simple OTG cable connected to a normal USB charging cable to connect the Due to the phone.
It only worked on my HTC Desire S with CyanogenMod on it (Android 2.3.7 I think), the Google phone with Android 4 just started charging.
I'm having some trouble in my application too, spontaneous packet loss that I haven't had the time to figure out yet..