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Absolutely; a parallelogram is a form of quadrilateral, in that it has four sides. If the four sides come in two sets of parallel lines, you have a parallelogram.
The answer depends on the shape of the quadrilateral and the form in which that information is given: for example, lengths of sides and angles, coordinates of vertices.
A rectangle is a 4 sided quadrilateral having two pairs of parallel lines of different lengths
All supplementary angles do not form a linear pair. The opposite angles of any quadrilateral inscribed in a circle (a cyclic quadrilateral) are supplementary but they are not a linear pair. However, all linear pair are supplementary.
Not necessarily. A linear pair of angles must be supplementary but supplementary angles need not form a linear pair. For example, the opposite angles of a cyclic quadrilateral are supplementary but they are (by definition) not next to one another.