Size:
The rule of thumb is 10 sq feet of outdoor space per standard size chicken and it is really the minimum since the chickens will peck and squabble if they can't move around.
Maintaining grass/ground cover:
The only way I can maintain grass is to have two runs so one can rest and recover while the other is in use. We have a small predator proof run 70 sq feet clad with 1/2" builders wire. The larger run is ~ 500 sq feet. It's fenced with 2"x2" cattle fencing on two sides and the wooden property line fence on the other two, Because of the open top and opened fencing we do get lots of wild birds in our run. If this will be a problem for you you should take it into consideration when you choose your fencing material. The hens are kept in the big run in the summer so that humans can use the yard, During the winter the hens use most of 1/2 acre while the run is blocked off to recover. Beginning in late September, while the hens free range in the bigger yard the run is raked out, tilled, mulched and seeded. I also add leaves as they fall.
Poop:
You can keep the poop down with thick ground cover since the hens will constantly scratch it under. It doesn't need to be the same all over the run. We use sand under the coop and, as the chickens tear out the seasonally planted fodder we plant I add wheel barrel of leaves that I much by going over them with the lawn mover. all fall and winter long. We have mainly pines on our property so we get our leaves from my neighbors. I bag the mulched leaves in garbage bags and keep them in the shed. The compost set up is in the large run so we walk and wheel barrel through it all winter. Sometimes by spring we run out of leaves and if the run gets muddy we add a bale or two of straw to the path, but generally I try to avoid straw since we live in an urban area and straw isn't east for the hens to scratch over completely so tends to get stinky and draw flies,
BTW I compose everything that I rake out of the coop and run along with kitchen waste and lawn clippings. Mixed 1/2 to 1/2 with native soil the the mixture of composed leaves, chicken poop, straw and sand makes a great planting medium for raised beds.