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YES to Question 1. For an arbitrary homomorphism $\phi\colon G\to F$ of connected algebraic groups over $k$, not necessarily affine, where $k$ is a $p$-adic field or $k=\mathbb{R}$, the image $\phi(G(k))$ is closed in $F(k)$. Proof. If $G$ is a connected $k$-group, and $X=G/H$ is a homogeneous space of $G$, then every orbit of $G(k)$ in $X(k)$ is open. ...


In the case of Jacobian varieties, the Weil pairing has an alternative (and beautiful) description in terms of Weil reciprocity for curves, and also in terms of Hilbert symbols. For elliptic curves, the Weil reciprocity part is an exercise in Silverman's book (#3.16). For general Jacobians (and both Weil reciprocity & Hilbert symbols) see Howe's paper ...

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