Let $k$ be a number field and $K/k$ be a finite Galois extension of degree $d=[K:k]$
with Galois group $\Gamma=\Gamma_{K/k}$.
Let $v$ be a place of $k$, and let $k_v$ denote the completion of $k$ at $v$.
Question 1. Is it true that there exists a canonical isomorphism
$$\nu_v\colon H^2\big(\Gamma,(K\otimes_k k_v)^\times\big)\stackrel\sim\longrightarrow \frac{1}{d_v}{\Bbb Z}/{\Bbb Z},$$
with $d_v=[K_w:k_v]$, where $w$ is a place of $K$ over $k$?
Now consider the adele ring $A_K=K\otimes_k A_k$, the idele group $I_K=(A_K)^\times$
and the idele class group $C_K=I_K/K^\times$.
There is a canonical isomorphism of class field theory
$$\nu\colon H^2(\Gamma,C_K)\stackrel\sim\longrightarrow\frac{1}{d}{\Bbb Z}/{\Bbb Z}$$
(is that correct?).
Consider the canonical homomorphism of $\Gamma$-groups
$$\phi_v\colon (K\otimes_k k_v)^\times\,\hookrightarrow\, (K\times_k A_k)^\times= I_K\longrightarrow C_K$$
and the induced homomorphism
$$\phi_{v,*}\colon\, \frac{1}{d_v}{\Bbb Z}/{\Bbb Z}\cong H^2\big(\Gamma,(K\otimes_k k_v)^\times\big)\longrightarrow H^2(\Gamma,C_K)\cong\frac{1}{d}{\Bbb Z}/{\Bbb Z}.$$
Question 2. What is the map $\phi_{v,*}\,$?
I will appreciate any references.
Best Answer
Let $G = \Gamma$, let $H$ be the decomposition group of $w|v$, so $H$ acts on $K_w$. Then
$$K \otimes_{k} k_v \simeq \mathrm{Ind}^{G}_{H} K_w,$$
and the same is true after passing to units. Hence, by Shapiro's Lemma,
$$H^2(G,(K \otimes_{k} k_v)^{\times}) \simeq H^2(H,K^{\times}_w)$$
But $H$ is identified with $\mathrm{Gal}(K_w/k_v)$, and the result you ask for is then a standard result in the computation of a Brauer group of a local field (Corollary 2, page 131, Serre's notes in Cassels-Frohlich). Note that Serre also shows that this group is generated by $u_{L/K} \in \mathrm{Br}(k_v)$ with invariant $1/d_v \in \mathbf{Q}/\mathbf{Z}$ ("canonically").
Actually, for a more direct reference, see Corollary 7.4(b) of Tate's notes in the next chapter of the same book on page 177.
As to the last question, the map is as you expect; viewing the first element as $1/d_v$ inside $\mathbf{Q}/\mathbf{Z}$ and the latter as $1/d$ inside $\mathbf{Q}/\mathbf{Z}$, it is induced by the identity map from $\mathbf{Q}/\mathbf{Z}$ to itself, so in your setting is the injective map sending $1/d_v$ to $(d/d_v) \cdot 1/d$. This is because $H^2(K/k,C_K)$ is more or less the $d$ torsion in $\mathbf{Q}/\mathbf{Z}$ which is the "final term" in the standard short exact sequence of class field theory:
$$0 \rightarrow \mathrm{Br}(k_v) \rightarrow \bigoplus_{v} \mathrm{Br}(k_v) \rightarrow \mathbf{Q}/\mathbf{Z} \rightarrow 0.$$
For a precise reference, combine the discussion in Consequence 9.6 of page 185 (of Tate's notes) with diagram (9) in section 11, bottom of page 195.
To make sure you look at the right edition of Cassels-Frohlich I am using the one with this page numbering: https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf