Yes.
A parallelogram is any 4-sided figure where the opposite sides are parallel. Only the special case where all four angles are 90° do you have a rectangle.
To visualize, picture a cardboard box, closed on top, and open on two sides. Move the top so that it is leaning a little, and both open sides are parallelograms.
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In fact, some are rectangles, but not all are.
True
Yes, all rectangles are parallelograms. However, not all parallelograms are rectangles.
All parallelograms are not rectangles. The angles of a parallelogram need not be right angles and so the figure need not be a rectangle.
No. Rhombuses that are also rectangles are called squares. Rhombuses are parallelograms with 4 equal sides, while rectangles are parallelograms where all the angles are right.