Is it possible to debug with Xcode through a Lightning Digital AV Adapter?
When I connect my iPhone 5 directly to my MacBook it shows up in Xcode.
When I connect the Lightning Digital AV Adapter to my iPhone, then plug the USB cable into the Lightning port on the AV adapter, Xcode does not detect my phone.
Is there no way to debug my app with Xcode while my phone is connected to an external display?
If the external display is over AirPlay, you can!
In the case of the Lightning AV adapter, the USB connection is charging only, it does not pass through data. Which is the crux of your issue.
There is a way to debug with an MFI device connected on older (non lightning) devices, I do not know about lightning connector devices but I would assume something similar will soon exist. For non-lighting devices this does work: CableJive dockStubz That provides a USB pass through before it the dock connector, which allows happy debugging time to occur.
There is some more information about the internals of your connector here: The Lightning Digital AV Adapter Surprise