In my part of the world..(Australia)...
All the words you mentioned are OK and are all used.
Chicken coop, henhouse, chicken house are used for smaller buildings where people keep chickens at home.
Barns and sheds are large buildings on chicken farms.
Battery hens are kept in small cages stacked in sheds. (very bad for the chickens)
Barn raised chickens are still cramped but they can move around.
Free range chickens can go outside and freely scratch for their food.
Eggs for sale show which type of conditions the hens were raised under.