raacampbell:
Does anyone know if connecting both USB and an external power supply is known to be problematic?
No, it isn't.
raacampbell:
If not, then I may have a separate problem with this circuit.
Clearly, if something smoked, you have connected something badly wrong.
No way we can tell what that was without clear pictures of the boards in question. Resolution 1024 by 768 maximum, a focused photo of that size (or less) will readily permit diagnosis, a blurry photo of any size is quite useless.