Any polygon with an even number of sides can have opposite sides that are congruent and parallel.
A four-sided polygon with opposite sides parallel and all sides congruent is a square.
Any polygon with 2n sides (n integer) where opposite sides are parallel, will have its opposite angles congruent.
There is no particular name. It need not be a regular polygon - for example, consider a stretched hexagon.
A four sided polygon where the opposite sides are parallel to each other (they must also by congruent). This also makes the opposite angles congruent. __ /_/ Sort of like that.
A square or a rectangle
A square is a four-sided polygon with two congruent parallel sides.
It cannot be a quadrilateral because if one pair of opposite sides are congruent and parallel, it is a parallelogram and so the other two sides are also (mutually) congruent and parallel. Any polygon with 5 or more sides will have an irregular form which satisfies these conditions.
A square or a rhombus