As KConrad says, the answer to the first question is "not yet known". For Iwasawa theory, you could consider the extension $L_\infty=\mathbb{Q}(\sqrt[p^\infty]{2},\zeta_{p^\infty})$ which is Galois over $\mathbb{Q}$ with Galois group isomorphic to $G=\mathbb{Z}_p^\times\ltimes \mathbb{Z}_p$ and some Iwasawa theory can tell you the structure (may be assuming that $2$ is a multiplicative generator $\pmod{p^2}$) of the $p$-part of the class group of your field, but nothing for $\ell\neq p$.
Already in the most basic situation where you consider the cyclotomic $\mathbb{Z}_p$-extension of the rationals, Iwasawa theory tells you that $p$ never divides the class number, and also that for all $\ell\neq p$ the power of $\ell$ dividing the class number of the $n$-th layer is bounded independently of $n$ (but may depend on $\ell$): this is a theorem of Washington. But it is still unknown whether only finitely many such $\ell$ do actually occur.