Do basic operators of RE (Union, Kleene star and Concatenation) have properties like associativity, commutativity, distrbutivity etc.? So in regular algebra we have some basic operations defined such as multiplication, addition, subtraction and division.

For these operations/operators, we have some properties like commutativity, associativity and distributivity.

Is there an analogue set of properties for the operations/operators on that we do with Regular Expressions?

I also believe that the set of all regular expressions together with its operations that are defined on the members of regular expression set form an algebraic structure similar to the integers and common operators that we learn about at arithmetics.
There are several known axiomatizations of the equational theory of Kleene algebra, see lecture notes from a course by Dexter Kozen. These axiomatizations give you a complete set of rules that can be used to deduce any identity involving regular expressions. The first ones were given in a classic paper of Salomaa, and others are also known.