No, only sometimes. If a parallelogram has four equal sides, then it's also
a rhombus, and could also be a square. Otherwise, it's just a parallelogram,
and has only two pairs of equal sides.
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Yes, and its opposite sides are always equal and its opposite angles are always equal. The angles and the sides can be any size.
A square. Four 90 degree angles allow a quadrilateral to have equal sides.
No. A rhombus has equal sides but is not regular because its angles aren't equal.
No. A quadrilateral is a polygon with four sides - of any length. A rhombus has four equal sides, a square has four equal sides AND four right angles. Incidentally, four equal sides for a rhombus implies two pairs of parallel sides and equal opposite angles, so those facts are not required to define a rhombus.
All four sides are aways equal in length.