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No.
They might, but they do not have to. A very long and narrow parallelogram would have diagonals that cross at a very sharp angle.no they donot
Yes * * * * * Usually not.
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.
Volume of a parallelogram = cross-section area times length