Interpreting some cohomology groups

Let $$C$$ be a smooth geometrically integral curve over a number field $$k$$, we do not assume $$C$$ to be proper, i.e., $$C$$ is not projective.

Under the spectral sequence $$H^p(k,H^q_{\mathrm{et}}(C_{\bar{k}},\mathbb{G}_m)) \implies H^{p+q}_{\mathrm{et}}(C,\mathbb{G}_m)$$, we will obtain the cohomology groups $$H^i(k,\bar{k}[C]^*)$$.

Question 1: How does one understand the elements of such groups, say for $$i=1,2,3$$?

In the case where $$C$$ is proper, we have $$\bar{k}^* = \bar{k}[C]^*$$, in other words, the invertible regular functions on $$C_{\bar{k}}$$ are simply the nonzero constant functions. Therefore, for $$i=1,2,3$$, we have respectively, $$0$$ (by Hilbert 90), Br$$(k)$$, and $$0$$.

Now let $$C_1$$ be the smooth completion of $$C$$. $$C_1$$ is now a smooth projective geometrically integral curve. The inclusion $$i : C \rightarrow C_1$$ induces the map $$i^*: H^1(k,\mathrm{Pic}(C_{1,\bar{k}})) \rightarrow H^1(k,\mathrm{Pic}(C_{\bar{k}}))$$.

Question 2: Are there any ways to explicitly describe $$i^*$$?