A paper airplane is affected by its shape in just about every way. Drag, efficiency and performance are all affected by the aircraft's shape.
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If drawn on a flat plane (paper, for instance), then yes, they are a 2D shape.
plane shape
A circle is not a plane shape. A circle is round a plane is not. A plane has parts that are similar to a circle shape.
No, but an octagon is.
A sheet of A4 paper could be classed as a plane 2D shape, as it has length x width. A shoebox is an example of a 3D shape, as it has length x width x depth.