Should I keep it grass or till it up? Up to you, but if it is long grass, trim it as long grass can lead to an impacted crop.
could i plant something there like pumpkins/squash/melons/Sunflowers? I have not had problems with the chickens eating the above mentioned plants because the leaves are scratchy, so they stayed away from them. However, don't let the chickens get to them until they are established...they will east the sprouting seedlings.
If i do will my hens destroy them? Depending on the plant, yes! My girls ate ALL of my broccoli, collards, and spinach. They try to eat as much of my grapes, blackberries, and strawberries as possible. They did leave my tomatoes, potatoes and peppers (bell and jalapeno) alone. A few strawberries they killed by scratching the dirt up and exposing the roots.
Can I put down straw or hay if the area becomes to wet or muddy or will it be fine? If it becomes too muddy, your plants may drown. If it is just the moisture from watering the plants and it isn't excessive (such as drip or small sprayers) you should be fine. If it is runoff from elsewhere or some massive sprinkler that soaks everything (including walkways), you may want to design flowline channels that divert excess water out of the area. Wet hay or straw will just breed mildew and attract harmful bugs (at least in our area anyway). Which leads to a quick point I should make...these are based off our experience, we aren't farmers (though we wish we were!), just have a small backyard ranch.
Are there any plants chickens shouldn't eat? Yes! Here's a quick list I found online.
http://www.poultryhelp.com/toxic1.html
