The answer to your first question is "no". In general, if $K/k$ is a cyclic Galois extension of odd order, then the order of the capitulation kernel (the subgroup of the class group of $k$ that dies when base-changing to $K$), is $[K:k]\cdot [\mathcal{O}_k^\times: N_{K/k}\mathcal{O}_K^\times]$. The second factor is the index in the integral units of $k$ of the subgroup generated by norms of units of $K$, and it can be non-trivial. See Iwasawa, A Note on Capitulation Problem for Number Fields for a concrete example of a field $k$ whose class group capitulates in a proper subfield of the Hilbert class field, using the above observation.

The answer to your first question is "no". In general, if $K/k$ is a cyclic unramified Galois extension of odd order, then the order of the capitulation kernel (the subgroup of the class group of $k$ that dies when base-changing to $K$), is $[K:k]\cdot [\mathcal{O}_k^\times: N_{K/k}\mathcal{O}_K^\times]$. The second factor is the index in the integral units of $k$ of the subgroup generated by norms of units of $K$, and it can be non-trivial. See Iwasawa, A Note on Capitulation Problem for Number Fields for a concrete example of a field $k$ whose class group capitulates in a proper subfield of the Hilbert class field, using the above observation.