Do basic operators of RE (Union, Kleene star and Concatenation) have properties like associativity, commutativity, distrbutivity etc.?
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.