is it possible to know if a language is regular if its equivalence classes are finite?
One of the main results of Myhill–Nerode theory is that a language is regular iff its Myhill–Nerode relation has finitely many equivalence classes. You can find a proof in many online resources. In particular, if all equivalence classes of the Myhill–Nerdoe relation of a language are finite, then it must have infinitely many equivalence classes, and so be non-regular.

The language $L^*$ is finite in two cases: if $L = \emptyset$ or if $L = \{\epsilon\}$. In any other case it is infinite.