I don't know of anything you can plant in your run that they won't kill. What they don't eat they scratch up. I think if you keep chickens in a run, you have bare dirt. I don't like any type of organic ground cover in the run, such as wood shavings, wood chips, pine needles, or straw. It can get wet and hold moisture and mold, which is bad for the girls. I think you are better off with the bare dirt. I do put bales of wheat straw inthe run for them to thresh and rip the bale apart so I can use it for mulch in my garden, but I do not leave it in there when it gets wet.
When will they lay? When they are ready and not until. It depends some on breed and some on the time of year. Each one is still an individual and will lay in her own time. It is possible that they can start to lay as young as 16 weeks, but not real likely. Usually, with maybe 10 dual purpose pullets of hatchery stock, I get my first egg around 18 to 20 weeks. Usually, by 23 to 24 weeks, about half are laying. Most have started by 27 to 28 weeks, but occasionally some can wait a whole lot longer. This is for chicks hatched in the spring so they reach laying age during the summer. Extreme heat can delay this or drop production too.