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The diagonals are equal and they bisect each other.
Yes.
Yes
Not always because the diagonals of a rectangle bisect each other but they are not perpendicular to each other.
The diagonals of any parallelogram (square, rhombus, rectangle, rhomboid) bisect each other. The difference is the the diagonals are equal in length for a square and rectangle, and not equal for a rhombus or rhomboid (oblique diamond).