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For (Q3), here is a class of structures whose VP is strictly intermediate in strength.

Theorem. The following are equivalent.

1. The class $L$ of constructible sets satisfies Vopěnka's principle. That is, any proper class of structures individually in $L$ has one elementarily embedding into the other.
2. $0^\sharp$ exists (see zero sharp).

Proof. $(1\to 2)$. Assume $L$ satisfies VP. Consider the structures of the form $\langle L_{\delta^+},\in,\kappa\rangle$. We get an elementary embedding $j:L_{\delta^+}\to L_{\gamma^+}$. This is known to imply that $0^\sharp$ exists. Thanks to a helpful discussion with Gunter Fuchs, here is an outline: let $\kappa$ be the critical point of $j$, and consider the induced $L$-ultrafilter $\mu$ on $\kappa$ defined by $X\in\mu\iff\kappa\in j(X)$. The ultrapower of $L_{\delta^+}$ by $\mu$ maps into $L_{\gamma^+}$ and hence is well-founded. To see that the full ultrapower of $L$ by $\mu$ is well-founded, consider any countably many functions $s_n:\kappa\to\text{Ord}$ in $L$. If $0^\sharp$ does not exist, we can cover this family of functions with a family of $\omega_1$ many functions in $L$. The union of the ranges of these functions has collectively size $\kappa$ altogether at worst, and so $L$ can find isomorphic copies inside $L_{\kappa^+}$, but the ultrapower of $L_{\kappa^+}$ by $\mu$ is well-founded. So the ultrapower of $L$ by $\mu$ is well-founded, and so we have a nontrivial elementary embedding of $L$ to $L$ and so $0^\sharp$ exists.

$(2 \to 1)$. Assume $0^\sharp$ exists and $\cal C$ 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 indiscernible parameters $\theta_0,\ldots,\theta_n$, and $A'\in \cal C$ is much further along, defined using indiscernibles $\theta_0',\ldots,\theta_n'$, in the same relative order, but much larger (although possibly some of them are the same), and plenty of room. Let $j:L\to L$ be an elementary embedding that arises by mapping $\theta_i\mapsto\theta_i'$ and other indiscernibles suitably. It follows that $j(A)=A'$, and that $j\upharpoonright A:A\to A'$ is elementary. So $\cal C$ satisfies VP. QED

Since $0^\sharp$ has intermediate consistency strength between ZFC and full Vopěnka's principle, this is an instance of (Q3).

The argument appears to generalize to the following:

Theorem. The following are equivalent, for any $x\subset\text{Ord}$.

1. The class $L[x]$ satisfies Vopěnka's principle. That is, any proper class of structures individually in $L[x]$ has one elementarily embedding into the other.
2. $x^\sharp$ exists.

For (Q3), here is a class of structures whose VP is strictly intermediate in strength.

Theorem. The following are equivalent.

1. The class $L$ of constructible sets satisfies Vopěnka's principle. That is, any proper class of structures individually in $L$ has one elementarily embedding into the other.
2. $0^\sharp$ exists (see zero sharp).

Proof. $(1\to 2)$. Assume $L$ satisfies VP. Consider the structures of the form $\langle L_{\delta^+},\in,\delta\rangle$. We get an elementary embedding $j:L_{\delta^+}\to L_{\gamma^+}$. This is known to imply that $0^\sharp$ exists. Thanks to a helpful discussion with Gunter Fuchs, here is an outline: let $\kappa$ be the critical point of $j$, and consider the induced $L$-ultrafilter $\mu$ on $\kappa$ defined by $X\in\mu\iff\kappa\in j(X)$. The ultrapower of $L_{\delta^+}$ by $\mu$ maps into $L_{\gamma^+}$ and hence is well-founded. To see that the full ultrapower of $L$ by $\mu$ is well-founded, consider any countably many functions $s_n:\kappa\to\text{Ord}$ in $L$. If $0^\sharp$ does not exist, we can cover this family of functions with a family of $\omega_1$ many functions in $L$. The union of the ranges of these functions has collectively size $\kappa$ altogether at worst, and so $L$ can find isomorphic copies inside $L_{\kappa^+}$, but the ultrapower of $L_{\kappa^+}$ by $\mu$ is well-founded. So the ultrapower of $L$ by $\mu$ is well-founded, and so we have a nontrivial elementary embedding of $L$ to $L$ and so $0^\sharp$ exists.

$(2 \to 1)$. Assume $0^\sharp$ exists and $\cal C$ 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 indiscernible parameters $\theta_0,\ldots,\theta_n$, and $A'\in \cal C$ is much further along, defined using indiscernibles $\theta_0',\ldots,\theta_n'$, in the same relative order, but much larger (although possibly some of them are the same), and plenty of room. Let $j:L\to L$ be an elementary embedding that arises by mapping $\theta_i\mapsto\theta_i'$ and other indiscernibles suitably. It follows that $j(A)=A'$, and that $j\upharpoonright A:A\to A'$ is elementary. So $\cal C$ satisfies VP. QED

Since $0^\sharp$ has intermediate consistency strength between ZFC and full Vopěnka's principle, this is an instance of (Q3).

The argument appears to generalize to the following:

Theorem. The following are equivalent, for any $x\subset\text{Ord}$.

1. The class $L[x]$ satisfies Vopěnka's principle. That is, any proper class of structures individually in $L[x]$ has one elementarily embedding into the other.
2. $x^\sharp$ exists.

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.)

For (Q3), here is a class of structures whose VP is strictly intermediate in strength.

Theorem. The following are equivalent.

1. The class $L$ of constructible sets satisfies Vopěnka's principle. That is, any proper class of structures individually in $L$ has one elementarily embedding into the other.
2. $0^\sharp$ exists (see zero sharp).

Proof. $(1\to 2)$. Assume $L$ satisfies VP. Consider the structures of the form $\langle L_{\delta^+},\in,\kappa\rangle$. We get an elementary embedding $j:L_{\delta^+}\to L_{\gamma^+}$. This is known to imply that $0^\sharp$ exists. Thanks to a helpful discussion with Gunter Fuchs, here is an outline: let $\kappa$ be the critical point of $j$, and consider the induced $L$-ultrafilter $\mu$ on $\kappa$ defined by $X\in\mu\iff\kappa\in j(X)$. The ultrapower of $L_{\delta^+}$ by $\mu$ maps into $L_{\gamma^+}$ and hence is well-founded. To see that the full ultrapower of $L$ by $\mu$ is well-founded, consider any countably many functions $s_n:\kappa\to\text{Ord}$ in $L$. If $0^\sharp$ does not exist, we can cover this family of functions with a family of $\omega_1$ many functions in $L$. The union of the ranges of these functions has collectively size $\kappa$ altogether at worst, and so $L$ can find isomorphic copies inside $L_{\kappa^+}$, but the ultrapower of $L_{\kappa^+}$ by $\mu$ is well-founded. So the ultrapower of $L$ by $\mu$ is well-founded, and so we have a nontrivial elementary embedding of $L$ to $L$ and so $0^\sharp$ exists.

$(2 \to 1)$. Assume $0^\sharp$ exists and $\cal C$ 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 indiscernible parameters $\theta_0,\ldots,\theta_n$, and $A'\in \cal C$ is much further along, defined using indiscernibles  $\theta_0',\ldots,\theta_n'$, in the same relative order, but much larger (although possibly some of them are the same), and plenty of room. Let $j:L\to L$ be an elementary embedding that arises by mapping $\theta_i\mapsto\theta_i'$ and other indiscernibles suitably. It follows that $j(A)=A'$, and that $j\upharpoonright A:A\to A'$ is elementary. So $\cal C$ satisfies VP. QED

Since $0^\sharp$ has intermediate consistency strength between ZFC and full Vopěnka's principle, this is an instance of (Q3).

The argument appears to generalize to the following:

Theorem. The following are equivalent, for any $x\subset\text{Ord}$.

1. The class $L[x]$ satisfies Vopěnka's principle. That is, any proper class of structures individually in $L[x]$ has one elementarily embedding into the other.
2. $x^\sharp$ exists.

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). Let $j:L\to L$ be the elementary embedding that arises by mapping $\theta_i\to\theta_i'$. 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.)

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.)