As to the trim being installed incorrectly, I don't agree. It is correct but not a good job at all. Go to the top of the coop and find the begining of the water leak. May even have to take the top board off. Once that is located make sure everything is bone dry and seal it. On all the out side wood I would use a lindseed oil or something similar to seal the wood. Then use a exterior caulking to caulk the horizontal board and vertical trim intersection joint. That should be done all the way down, top to bottom. The meeting of the boards and trim should be cut so tight that a caulking wouldn't be necessary. But yours had some pretty good size gaps so caulking is going to be necessary. Also don't just work on the one corner, do the whole coop the same way. Then you'll have a water tight coop. Remember old porous wood will act like a sponge around water. You'll find it does the same with sealer.
One other thing I would look at is the roof above the corner. Be sure it's not leaking and allowing water to run down and causing the problem. Always start on the top and work you way down looking for the source of the problem.
Also as has been said, use a good exterior paint on it too. Don't forget the bottom either. It needs care.
One other thing I would look at is the roof above the corner. Be sure it's not leaking and allowing water to run down and causing the problem. Always start on the top and work you way down looking for the source of the problem.
Also as has been said, use a good exterior paint on it too. Don't forget the bottom either. It needs care.