Yes, in theory, the scale will register the weight of the helicopter, assuming all of the air put in motion by the rotors will impact the scale, and assuming the helicopter is not moving vertically.

In practice, it won't register the total weight, because there are losses to the fluid momentum (the air), because it disturbs the surrounding air, creating vortices and also because of the viscosity of the air resulting in heat loss. In addition, the lift varies spacially as the roters spin, so there will be fluctuation on the scale readout. But, neglecting these losses, and taking an average of the reading, the scale will register the weight of the helicopter.

If the helicopter is climbing, the scale will register more, if decending then less.

And yes, the birds add to the weight of the airplane, even if they are flying.