There is no specific name, except in the case of a triangle: scalene.
You should not call it simply an irregular polygon, because a regular polygon implies that ALL sides are equal and ALL angles are equal. So a rhombus, where all four sides are equal, is still irregular.
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It is an IRREGULAR shape.
a quadrilateral
a shape with 5 equal sides is a pentagon
this shape is called a trapizoid see heres a picture!
That depends on what you mean by "diamond shape." In a general shape, no, the sides need not be exact. If the shape is distinctly called a "rhombus," then mathematically speaking, yes, all sides must be equal, per the definition of a rhombus.