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Yes, the opposite angles in a regular quadrilateral are equal.
No. For example, if one angle measures 100 degrees, and its adjacent angle is 80 degrees, then the opposite angles would be either 200 or 160 degrees, but in order for a quadrilateral to be inscribed in a circle the opposite angles would have to equal 180 degrees. A parallelogram can be inscribed in a circle if it is a rectangle.
Diagonally opposite pairs of angles will always be equal, but the only time all angles in a quadrilateral parallelogram are equal is when that shape is a rectangle.
It is not possible for a quadrilateral in Euclidean plane geometry to have no equal angles and still have its opposite sides parallel.It's possible for a quadrilateral to have no equal angles and two of its sides parallel (opposite ones, obviously; adjacent sides can't possibly be parallel). That would be a trapezoid.
rhombus. The rhombus is a quadrilateral with all sides equal in length. It is also a parallelogram, so opposite sides are parallel and equal in length, and opposite angles are equal.