YES
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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.)
No but all parallelograms are trapezoids
No, but all parallelograms are trapezoids.
No. All rhombi (rhombuses) are parallelograms but all parallelograms are not rhombi.
Yes, they will all tessellate an infinite plane.