It might not happen right away, but the chickens will take care of your "grass issue" for you.
The coop looks nicely built, but I definitely wouldn't exceed 4 birds (and it might max out at 3 depending on the actual space in the interior of the enclosed area) but it appears to be chicken wire on the run. If predators such as dogs, raccoons, foxes, etc. are a concern you should consider adding hardware cloth or welded wire to reinforce it.

The coop looks nicely built, but I definitely wouldn't exceed 4 birds (and it might max out at 3 depending on the actual space in the interior of the enclosed area) but it appears to be chicken wire on the run. If predators such as dogs, raccoons, foxes, etc. are a concern you should consider adding hardware cloth or welded wire to reinforce it.