A quadrilateral doesn't have to have any pairs of parallel lines. It depends what kind of quadrilateral you are talking about...
Rectangle- Yes (2 pairs)
Square- Yes (2 pairs)
Trapezoid- Yes (1 pair)
Parallelogram- Yes (2 pairs)
Rhombus- Yes (2 pairs)
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A rectangle, by definition is an irregular quadrilateral. It has two pairs of parallel sides.
That depends on witch quadrilateral your talking about.
A diamond has two pairs of parallel lines.
A quadrilateral is any figure with 4 sides. ("quad" = 4, "latus" = side). It doesn't have to have any parallel lines. If it happens to have two sides that are parallel, then it's a trapezoid. If it happens to have two pairs of parallel sides, then it's a parallelogram. Common parallelograms are rectangles, squares, and kites.
A parallelogram has 2 pairs of parallel lines