Square
the answer is a square * * * * * No. It is equilateral. An equilateral triangle, or a rhombus are shapes that have equal sides, but neither are a square. The answer is not a regular polygon since that also requires equal angles. A rhombus is equilateral but it is not equiangular and so not regular.
Square
there are several names for rhombus quadrilaterals. rhombus quadrilateral is the main name of a rhombus quadrilateral. some other names arent very similar to this name but can still describe it.
A diamond is not a square, but a square is a diamond. A diamond is another name for rhombus.
The special name is rhombus.
square
the answer is a square * * * * * No. It is equilateral. An equilateral triangle, or a rhombus are shapes that have equal sides, but neither are a square. The answer is not a regular polygon since that also requires equal angles. A rhombus is equilateral but it is not equiangular and so not regular.
Square
There is no special angle and so it does not have a specific name.
Yes, the common name for an equiangular equilateral quadrilateral would be a square, a special instance of a rhombus (four sided figure with equal sides) which is a common interpretation of "diamond" in two dimensions. If by diamond you mean "kite" then no, by definition kites are not equiangular.
A Rhombus
Any quadrilateral that is not a rhombus (or square) although there is no specific name.
Rectangle.
Rhombus is the geometric name.
Rhombus, or square if there are any right angles
the name rhombus came from German languages
I don't use the name "rhombus-quadrilateral", but perhaps the answer to your question is that a rhombus is a special kind of parallelogram