I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a number field.

• Question 1. Is the quotient of $\widehat{\mathcal{O}}_K^{\times}$ (the units in the profinite completion of $\mathcal{O}_K$) by the closure of the unit group $\mathcal{O}_K^{\times}$ of $\mathcal{O}_K$ isomorphic, via the Artin map, to the Galois group of the maximal totally real sub-field of the maximal abelian extension $K^{\rm ab}$?
• Question 2. Is the Galois group $\text{Gal}(K^{\rm ab}/H)$, for $H$ the Hilbert class field, isomorphic to $\widehat{\mathcal{O}}_K^{\times}$ modulo the closure of the global units? (this would be in conflict with the first question, so I am also implicitly asking how to tweak this statement). Can it be at least written as a quotient of $\widehat{\mathcal{O}}_K^{\times}$ modulo the closure of some subgroup related to $K$ and $H$?

As a side question, I know that when $K$ has class number one, the idéle group is $\mathbf{A}_K^{\times} = (K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$. Is there a description of $\mathbf{A}_K^{\times}$ in general, involving $H$? (maybe as an extension of topological groups, involving the direct product $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$ as one of the terms)