1. All angles are right-angles.
2. All lines are parallel.
3. 2 pairs of 2 lines are the same length.
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The essential properties are that a rectangle is a quadrilateral whose sides met at right angles. There are many other properties but all of them follow from this.
I don't think there's a three dimensional rectangle, and if the two dimensional triangle exists in three dimensional space, it would have strange properties because it only has 4 edges. There however, a three dimensional rectangular prism. It has 12 edges.
Yes. There is a shape that has all the properties of a rectangle and all the properties of a rhombus at the same time. It is called a square.
A rectangle is a parallelogram with angles of 90 degrees. A square has those properties (and more) so it is a rectangle.
For a start, a square IS a rectangle, so it has ALL the properties of a rectangle. A square has the additional property that all of its sides have the same length.