Bird flying in a cage

If the cage is completely closed, it doesn't make a difference if the bird is hovering inside it or if it sits on the ground. When flying, the bird pushes air to the ground which will exert a downward force on the cage exactly equal to the weight of the bird. This is a direct consequence of the conservation of momentum and Newton's second & third law. Since no additional external force is acting on the cage-bird-system when the bird is flying as compared to when it's not, the acceleration on the cage can be no different. The effect due to the flying bird only concerns the internal forces and since action=reaction, they cancel.

However, if the cage would not be closed, some of the 'wind' due to the bird could escape the cage and would become an external force, making the cage-bird-system lighter.

To answer the second question first, a completely closed cage is a system which comprises the exterior cage, the air inside and the bird. So what all forces do we need to balance now? Gravity on air, bird and the cage. OK, So can the bird apply an additional force if it sits or swings or flap around the walls of the cage? And the answer might be against intuition but it is NO!

Let's imagine if it was a 'yes'. If the bird could apply a force on it's own closed system (or internal force) it could accelerate its own system as long it desires. Now this isn't right, is it? Otherwise space ships could just propel themselves by their travelers constantly kicking on the inside!

The air inside the cage does two things it expands and contracts (changes pressure) when the bird moves around or flaps its wings. So when the bird pushes some air towards the bottom of the cage the cage gets a force (a small impulse) which is immediately redrawn by another force now, on the top of the cage, by the elastic air that obeyed Newton's third law. In fact the bird itself will feel this recoil (that's why a bird can fly) and if the flap was strong enough it will hit the cage's top and does the same as above. So all you'll feel is an up and down movement of the cage! This is just a rearrangement of masses inside the cage, but no net forces are acting on the center of mass.

Now, for an open system external forces comes in to action. To put it simply, the bird will draw some air to the inside and this new air will apply new force that you will feel as a pull (since the air could now escape through the openings, the case above is not valid!)

Buoyant force has nothing much to do here except that it is what slightly reduces our weight (or the bird's) due to pure gravity by pushing us upwards since we are displacing the medium (air) we are in (Archimedes, eureka moment!) Thus the bird need to apply a force slightly less than it's gravitational force by earth in order to fly, but as I said no big deal here.

Now all this would get even more interesting by imagining a balloon instead of a bird. Think about it and tell me what you think.