Yes. If a pair of opposite sides are not equal,
then you don't have a parallelogram.
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Parallelograms (if the equal sides are opposite one another) or kites (if they are adjacent).
Parallelograms have two pairs of parallel sides, trapezium only one. Both pairs of opposite sides of a parallelogram must be equal, in a trapezium at most one pair can be. Opposite angles of a parallelogram must be equal, not so for a trapezium.
parallelograms, and rectangles (parallelograms)
No. In a parallelogram, both pairs of opposite sides are parallel.
No trapezoids are not parallelograms. Trapezoids are actually quadrilaterals and they have one pair of opposite sides. Parallelograms by definition have 2 pairs of parallel sides.