Answer: Lawn grass needs direct sun all day or for at least half the day, so it does not sound like grass will be a viable option for your shady yard. You might try growing a shade tolerant groundcover such as Vinca minor, Ajuga reptans, Liriope spicata (creeping lilyturf) or English ivy (Hedera helix). All of these are quite resiliant when it comes to traffic. Although I have not seen them tried with chickens, these plants are quite sturdy and I think they would have a pretty good chance of standing up to and recovering from the occasional visit.
When you plant, loosen the soil and work in a generous layer of compost to give the groundcover an improved rooting environment, keep them weeded and watered while they become established, and also mulch with organic mulch in between the plants to help the soil stay naturally moister until they fill in.
As far as the Kudzu, your ground cover simply will not be able to outcompete this weed. You may find the following information helpful. Please check with your local county extension for the most up to date control recommendations, including an possible additional chemical controls available to homeowners in your state. (The article lists only products containing glyphosate.)
Good luck with your project!
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