Only if it is a square.
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No. In fact, a rhombus cannot be cyclic - unless it is a square.
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
A rhombus may be a square or just a rhombus (a rhombus is merely called a rhombus when there are no 90 degree angles).
Rhombus Not a SquareIn a square, the sides are perpendicular. In a rhombus, they aren't. A rhombus is never a square.