Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$
Question 0. Applying the etale cohomology functor $H_{et}^i(k,-)$, I know that $H_{et}^i(k,\mathbb{G}_m) = H^i(k,\bar{k}^*)$, where the latter is a Galois cohomology group. What is the Galois cohomological equivalent for $H^i_{et}(k,\mu_n)$?
Let $\mu_\infty := \mathrm{colim}_n\mu_n$, this group can be interpreted as the set of all $n$-th roots of unity, i.e., $\mu_\infty \cong \mathbb{Q}/\mathbb{Z}$. We work only in Galois cohomology.
Question 1. I would like to know if we could obtain a similar Kummer sequence involving $\mu_\infty$ instead of $\mu_n$ for a fixed $n$. The reason for this is that I came across an exact sequence $$0 \rightarrow \mathrm{Br}(k) \otimes _\mathbb{Z} \mathbb{Q}/\mathbb{Z} \rightarrow H^3(k,\mu_\infty) \rightarrow H^3(k,\bar{k}^*) \rightarrow 0$$ but I am unable to derive it from the usual Kummer sequence.
Question 2. The idea for this exact sequence is to prove that the middle term is 0. It is known that we have $H^3(k,\bar{k}^*)=0$ in our setting and for any non-archimedean place $v$, we have $\mathrm{Br}(k_v) \subset \mathbb{Q}/\mathbb{Z}$. But I'm also unsure how does it imply that $\mathrm{Br}(k) \otimes _\mathbb{Z} \mathbb{Q}/\mathbb{Z}= 0$.