The shape and size of a bird's beak would most likely be affected by what environmental limiting factor ?

By the availability of food in plain open. If the main source of food for a certain bird species would be hidden safely in small and hard-to-reach places, then the bird beak would look differently over successive generations, compared to those where food would be easily found on the open. 

Used to water and cool climates, probably not thick coated or fluffy. they most likely have thin coats and have to survive to find food. sorry if this wasn't the answer you were looking for..: )