It's been a while since I lived there full time, but in general:
Cedar Grove is a "separate" place.
Edgewood is now technically an incorporated town with a mayor etc.
The post office for folks over an area of about 200 square miles is located in Edgewood, so that's why someone who lives in Cedar Grove, Venus, Barton, Bachelor or Sedillo might have an Edgewood mailing address.
I don't know if Edgewood has hired any cops yet.
Last I heard, Cedar Grove had not incorporated.
Try
www.townofedgewood.com for more info.