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](http://en.wikipedia.org/wiki/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.