The most famous of all 4 sided quadrilaterals is the 'Golden Rectangle' whose dimensions are in the ratio of about 1 to 1.618
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You could call it a regular quadrilateral, or an equilateral rectangle or even an equiangular rhombus, if you prefer.
I am not sure what he question means. A quadrilateral can be named by identifying its four vertices. This can be done by assigning a different letter (from any alphabet), number or other character to each vertex. In so far as the set of such naming devices is infinite, there are infinitely many ways to name a quadrilateral.
It does not have a specific name. I think that, of the named quadrilaterals, only a trapezium can have four different angles.
It is an inscribed quadrilateral or cyclic quadrilateral.
It is simply a quadrilateral - there is no special name for such a quadrilateral.