Polynomial with icosahedral symmetry

No, there are no invariants of smaller degree.

Let $G = PSL_2(\mathbb{F}_5)$. Assuming that you are working over a field of characteristic $0$ (in fact, it would be enough to work over a field of characteristic not dividing $\# G$), it is a theorem of Klein, proved in his book on the Icosahedron, that the invariant ring $K[v,w]^G$ is generated by the polynomials \begin{gather*} v^{11} w - 11v^6w^6 - vw^{11}\\ v^{20} + 228 v^{15} w^5 + 494 v^{10} w^{10} - 228 v^5 w^{15} + w^{20} \\ v^{30} - 522 v^{25} w^5 - 10005 v^{20} w^{10} - 10005 v^{10} w^{20} + 522 v^5 w^{25} + w^{30} \end{gather*}

This can be proved by computing the Poincare series using Molien's formula (or by computer algebra e.g., Magma's FundamentalInvariants).