Maybe make several picnic tables to go with your patio umbrellasCould you fashion a large table in the run abutting the coop.
I think the wire top is a good idea--it will absolutely keep the chickens from ever flying into the neighbors yard, and it will keep things like hawks, owls, and cats from getting into the run.
Putting the umbrellas inside the run is a good idea, too. That way they cannot blow away into someone else's yard. (They could blow against the inside of the run and get damaged, so you would still want to put them away if you're expecting heavy wind, but at least they won't go anywhere.)
For fast-growing vines, a few possibilities are cucumbers, squash or pumpkins (vining type, not bush type), some varieties of tomato plants, peas or beans (the tall or climbing varieties).
For longer-lived vines, I've heard that grapes are good too.