No, If all angles are equal and all sides are equal, then it is a regular polygon (otherwise it is 'irregular'). In this case, parallelogram is irregular.
as far as I know a polygon is anything with more than 3 sides. So technically yes.
A square is always a parellelogram since it has equal sides.
A rhombus. More information... The no right angles is throwing me, but I will add this comment: A square A four-sided polygon having equal-length sides meeting at right angles. The sum of the angles of a square is 360 degrees. A Parellelogram is a four-sided polygon with two pairs of parallel sides. The sum of the angles of a parallelogram is 360 degrees. A Rhombus is a four-sided polygon having all four sides of equal length. The sum of the angles of a rhombus is 360 degrees. So, a Square is actually a Parellelogram and Rhombus as well; and it has 4 right angles. So... I am not sure that Parellelogram and Rhombus are what you were looking for,