For (Q3), here is a class of structures whose VP is strictly intermediate in strength.
Theorem. If $0^\sharp$ exists, then every proper class of constructible structures, that is, any ${\cal C}\subset L$, satisfies Vopěnka's principle.
Proof. Suppose that $0^\sharp$ exists and that $\cal C$ is a proper class of structures, with ${\cal C}\subset L$. So each element of $\cal C$ is defined in $L$ by some formula $\varphi$ using some ordinal indiscernible parameters. By going to a subclass, we may assume that they are all defined using the same formula. Suppose that $A\in\cal C$ is defined by $\varphi$ using indiscernibles $\theta_0,\ldots,\theta_n$. Pick some $A'\in\cal C$ defined using indiscernibles $\theta_0',\ldots,\theta_n'$, in the same relative order, but larger (with possibly some of them the same), with plenty of room. Let $j:L\to L$ be an elementary embedding that arises by mapping $\theta_i\to\theta_i'$ and extending to other indiscernbles. It follows that $j(A)=A'$, and that $j\upharpoonright A:A\to A'$ is elementary. So $\cal C$ satisfies VP. QED
Meanwhile, the assertion that any proper subclass of $L$ is strictly weaker than full VP, if they are consistent, because VP is much stronger than $0^\sharp$ in large cardinal consistency strength.
And it is not provable outright in ZFC, as explained in your answer to Chris's question.
(Posting my answer now, it seems to me that $0^\sharp$ is likely simply equivalent to the assertion that $L$ satisfies VP; I'll give this some more thought and edit later if this seems right.)