Actually, you are 100% on the right track.
Coarse wood chips, the sort you get from a tree-trimming service, are usually considered the gold standard for managing mud and odor.
Yes.
Likewise diversion ditches to move water around the chickens' area. But there has to be somewhere for the water to go and neighbors don't appreciate it being sent into their yards.
Building up the ground level with layers of dry organic material is probably the best option unless there is a clear place to send the water.
"First, fix the drainage," is my usual advice, but it's not always feasible on a small city lot.