Let $C$ be a smooth projective curve of genus $\geq 1$ over a number field $k$ with a $k$-rational divisor of degree $1$ inducing the embedding $C \hookrightarrow J$, where $J$ is the Jacobian variety of $C$. I know that $J$ corresponds to the divisor classes of degree zero of $\bar{C} := C \times_k \bar{k}$ and so we have the identification $J \cong \mathrm{Pic}^0(\bar{C})$.
Question 1. In the proof of Lemma 2.4 of this paper, it is given that the map $$H^1(k,\mathrm{Pic}^0(\bar{J})) \rightarrow H^1(k,\mathrm{Pic}^0(\bar{C}))$$ of (etale) cohomology groups is an isomorphism. Why is this the case? Do we actually have $\mathrm{Pic}^0(\bar{J}) \cong \mathrm{Pic}^0(\bar{C})$?
By the following commutative diagram, it follows from the isomorphism of the bottom map that $H^1(k,\mathrm{Pic}(\bar{J})) \rightarrow H^1(k,\mathrm{Pic}(\bar{C}))$ is surjective.
$$\begin{array}[c]{ccc} H^1(k,\mathrm{Pic}^0(\bar{J}))& {\rightarrow}&H^1(k,\mathrm{Pic}(\bar{J}))\\ \downarrow &&\downarrow\\ H^1(k,\mathrm{Pic}^0(\bar{C}))& {\rightarrow}&H^1(k,\mathrm{Pic}(\bar{C})) \end{array}$$
Now suppose $C$ is projective with one singularity that is a multiple point, then $\mathrm{Pic}^0(\bar{C})$ is an extension of $\mathrm{Pic}^0(\tilde{C})$ by an algebraic group reflecting this singularity of $C$, where $\tilde{C}$ is the normalization of $\bar{C}$. Here $\mathrm{Pic}^0(\bar{C})$ is called the generalized Jacobian variety. I believe the algebraic group in this case is $\mathbb{G}_m^{r-1}$, where $r$ is the degree of the multiple point.
Question 2. Let $X$ be the smooth affine curve defined by removing the multiple point of $C$. One has an embedding by Serre $X \hookrightarrow \mathrm{Pic}^0(\bar{C}) = :S$ which is analogous to $C \hookrightarrow J$. Do we also have the surjectivity of $H^1(k,\mathrm{Pic}(\bar{S})) \rightarrow H^1(k,\mathrm{Pic}(\bar{X}))$?
EDIT. I've change question 2 slightly, because of a misconception explained by the comments below. In this case, we can relax the condition that genus of $C$ is at least 1, since for genus zero it will be trivially true.
