Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective family of curves with smooth generic fiber. The special fiber has at most nodal singularities. Then,

1) Considering the morphism $\mathrm{Pic}(X_R) \to \mathrm{Pic}(X_k)$ as a morphism of schemes, are the fibers always irreducible? Are they of dimension $h^1(\mathcal{O}_{X_k})$? (the later dimension count is my guess using a torsor action argument from deformation theory)

2) Can the specialization map $\mathrm{Pic}(X_K) \to \mathrm{Pic}(X_k)$ be interpreted as a morphism of schemes (rather than just as groups)? If so, are the fibers irreducible and of what dimension?

Any reference/hint in this direction will be most welcome.

NB. If necessary assume that $X_R$ is regular.