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It depends on the rotation of the square. If it is aligned to the X and Y axes, then there are two horizontal lines. If it is not aligned to the X and Y axes, there are no horizontal lines.
It must be. A cylinder is described by the volume swept by a rectangle in rotation about one of its axes. The rotation will create a circle on the opposite ends.
It depends on the pyramid. If it is a square based pyramid, a horizontal plane will give a square cross section, a plane inclined by a rotation parallel to one of the base axes will give a rectangular cross section whereas a plane inclined by rotation along both basal axes will result in a parallelogram cross section. Not sure how you get a parallelogram from a pentagonal or hexagonal (etc) pyramid.
Axes is plural; axis is singular (only one of them).
The x and y axes on the Cartesian plane intersect at the origin at right angles