Some of what I see in the pictures looks like rooster damage. If you are sure that you don't have a roo, then you may have a dominant hen that is doing mounting behavior. It sometimes happens, especially if there is no roo present in the flock. Some observing will tell you if that's going on. The bare backs above the tail, on the shoulders, and back of heads are all places that commonly lose feathers when they are mounted roughly. The bare butts are often feather picking, especially on the roosts. The dimensions of your coop sound ok, depending on your roosting layout and space, however I don't think your run is large enough for 30 birds if that is all the space they have 24/7. Over crowding and boredom are the most common causes of feather picking. Recommended space sizes are usually minimums, and the actual behavior of the birds themselves is what really matters. A space that works for one group of birds may be much to small for another group, it's very dependent on the individual birds themselves, their temperaments and personalities. One dominant bird can make a space too small. Pictures of your coop, roosting layout, and of the attached run might be helpful if you want idea's or advice on that.
