Assume that the Tate-Shafarevich group $\operatorname{Sha}(K,E)$ is finite.  For each element of $\operatorname{Sha}(K,E)$, take the corresponding genus $1$ curve, and let $V$ be the product of all of them.  Then the field version of your first question is exactly asking for a field extension $L$ such that $V(L)$ is nonempty.  As you say, there are many such $L$, but is there a canonical such field $L$?  Yes, let $L$ be the function field of $V$!  (Admittedly, this is rather tautological, but I'm not sure that one can do much better if you really want a canonical answer.)

As for the "norm map", it is perhaps more commonly known as the corestriction map in Galois cohomology, $\operatorname{Cores} \colon H^1(L,E) \to H^1(K,E)$, which can be restricted to the locally trivial elements if you want.  As surely you know already, the fact that $\operatorname{Cores} \circ \operatorname{Res}$ equals multiplication by $[L:K]$ is responsible for the generalization of what you called Heegner's lemma, that if an element of order $n$ in $H^1(K,E)$ restricts to $0$ in $H^1(L,E)$ for some finite extension $L$ of $K$, then $n$ divides $[L:K]$.

What I find more amusing is the following:  Let's continue to assume finiteness of Tate-Shafarevich groups.  Let $X$ be a genus $1$ curve corresponding to $x \in \operatorname{Sha}(K,E)$ of prime order $n$; then there is some $y \in \operatorname{Sha}(K,E)$ such that the Cassels-Tate pairing $\langle x,y \rangle$ gives $1/n$ in $\mathbf{Q}/\mathbf{Z}$.  For every degree-$n$ extension $L$ of $K$ such that $X(L) = \emptyset$ (i.e., the restriction $x_L:=\operatorname{Res}_{L/K}(x)$ is nonzero), there is a *new* element $z \in \operatorname{Sha}(L,E)$ such that the Cassels-Tate pairing $\langle x_L,z \rangle$ for $L$ gives $1/n$: this $z$ has to be new, i.e., not just $y_L$, because of how the Cassels-Tate pairing behaves under field extension: $\langle x_L, y_L \rangle = [L:K] \langle x,y \rangle = 0$.  So we expect to get elements of $\operatorname{Sha}(L,E)$ for various $L$ popping up all over the place!