A dog run doesn't have to be a cage but separation is sometimes necessary especially with a multiple-dog household.
You don't have to use chemicals and herbicides to have a nice green lawn. There are weeds in this photo of my lawn but I hand pull, use organic solutions and I mow at a 3.5" height.
Using more native plants attracts birds that help keep some insect population down and lessen the need for insecticides.
Even in the winter some dogs can stay out all season ;-). These are winter containers with a little extra.