Are BMX threaded headsets and other threaded headsets interchangeable? I want to buy a threaded headset for an old road frame that takes 1 inch threaded steerer tube / headsets. I see headsets for sale online that are labeled as "1 inch threaded BMX headset". Is there any compatibility issue with trying to install it on a road bike? What about non-BMX-labeled headsets being installed on BMX frames? I figure I can get a headset a little bit cheaper if I could use one that is labeled as "BMX".

No, they're not interchangeable. BMX 1" threaded headsets have an outer cup diameter of 32.7mm - they wouldn't fit in a standard road frame. Standard (ISO) threaded headsets have an outer cup diameter of 30.2mm - they'd fit loosely in a BMX frame.

Sheldon Brown's Bicycle Headset Dimension Cribsheet is a useful reference for headset interchangeability.