Yes. Remember that rectangles and squares are both parallelograms. Square tiles are the most common. The next most common are rectangles. Both shapes can tessellate finite planes.
Other parallelograms can tessellate infinite planes (or finite ones if you allow parts of the parallelogram along the edge of the plane.)
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They can tesselate - provided they are the correct shapes.
No
Yes. Can it tesselate R^3? Only if its base pentagon is non-regular.
there are loads. ones that do tesselate are hexagons, squares and triangles but none of the others tesselate
No, you cannot.