= Solution
For a finite Galois extension $L/K$, the kernel of
$$
E(K)/nE(K)\longrightarrow E(L)/nE(L)
$$
is finite. Indeed, if $P=nQ$ for $Q\in E(L)$, then $\sigma\mapsto\sigma Q-Q$ is a cocycle in the finite $G$-module $E[n](L)$, and the resulting map from the kernel to $H^1(G,E[n](L))$ is injective.
The analogue of part b assumes $E[n]\subset E(K)$. Given $P\in E(K)$ and $Q\in E(\overline K)$ with $nQ=P$, every conjugate of $Q$ is $Q+T$ for some $T\in E[n]$. Hence $K(Q)/K$ is Galois and
$$
\operatorname{Gal}(K(Q)/K)\longrightarrow E[n],
\qquad
\sigma\longmapsto\sigma Q-Q
$$
is an injective homomorphism. These are the two elliptic forms of the <Kummer pairing>.
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