OK, here is what I do to hard boiled eggs to get them to come out of the shell easier & be the right color. I bring the water with the eggs in to a boil. Set a timer, lower heat to just a soft boil, and let boil for 12 minutes for large eggs, less for smaller (10 min) . Then remove from stove & put in cold water and keep the water cold. When I can handle them, I crack them so the shell has several cracks in it as if I was going to peel them at that time.l I let them sit in the cold water. The cracked shells allow the water to seep into the egg between the membrane & the egg contents. After an hour or so, I never timed this so it might be longer, anyways when the eggs are good & cold, I shell them. Usually the shells peel off easily. Once in a while one stays tough to peel. If you leave them in the refrigerator overnight, they peel even easier.
The green yolk is due to overcooking.
Hens poop in the nest usually because they roost on the edge of the nest box if they are raised up. If I have a poopy nest box, I'll either put it on the floor, or make it lower than the roosts are, or if they still roost on the sides, of course, they always roost with their butts over the box, not the other way around, I try to use a box that they can't stand on the edges. They are hard to find, so mostly I just dig out the poop every morning.