Most interesting reading!
As far as I recall from my vet anatomy days, air passes into the lungs first and then on to the air sacs. When the bird breathes out, some air passes straight from sacs to trachea but most travels from sacs to lungs and then out. Birds have no diaphragm, and both inspiration and expiration are active processes that the bird performs by lifting and lowering its sternum. Hence gentle handling is necessary - as any sustained pressure on the breastbone will suffocate the bird.
Birds' nostrils are quite functional. Mouth breathing is abnormal and most commonly signifies disease/parasites/heat stress. Firmly taping shut the beak of a bird (e.g to protect hands during examination!)does not lead to any respiratory distress.