(1) Irreducible representations are more about understanding representations than they necessarily are about understanding groups. You said it yourself: they are about decomposing representations into their constituent parts, like natural numbers being factored into products of prime numbers, or molecules decomposed as a collection of atoms.

By themselves groups are just sets with binary operations satisfying some properties - and from this vantage point alone that doesn't inspire much confidence in their interestingness, since in a certain sense 'most' algebraic structures are boring and unimaginably complicated - but the defining properties of the group operation are abstracted from the idea of *symmetry*, and this motivation is resurrected in the idea of a *group action*. It is debatable to what extent it is an exaggeration to say the whole point of groups is to act on things. Since space is an important concept, in particular vector spaces, it makes sense to study linear group actions in their own right, i.e. representations.

That said, it's true that representation theory has been applied to group theory, combinatorics, quantum physics, chemistry and other areas. For instance, see

(BTW there are, actually, naturally occuring groups which are not just symmetry groups:

(2) The phrase "irreducible representation of each group element" makes no sense to me. Perhaps you could make this question more intelligible. If you mean find the value of $\rho(g)$ for all $g\in G$, where $\rho:G\to\mathrm{GL}(V)$ is an irreducible representation, then these elements are only defined up to pointwise conjugation of $\rho$ by linear transformations. Generally, the work involved pertaining to irreducible representations is finding ways to describe them, classify them, and calculate their characters (since most things you want to know about representations may be calculated practically using just the character table).