I'm confused by this new feature. Apparently iOS 10 and the 7Plus can shoot RAW files, though the camera app doesn't support doing so and you need a 3rd party camera app to shoot in RAW? Is this correct?
Yes you need a 3rd party camera app to shoot raw as all Apple did was give developers an API to enable RAW shooting.
Editing the resultant RAW file can either be done on your computer with LightRoom or you can do it on your phone with LightRoom Mobile(requires a subscription) or Snapseed (free).