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.
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Yes.
No, you cannot.
You would cut off a corner.
No
Well, a parallelogram can tile. Take one vertex and make sure the parallelogram can go around 360 degrees.