I'm talking specifically about instantons on four-manifolds, but my confusion here is probably of a more general nature. So I'd also appreciate less specific answers!
Okay, so I know that in physics, if you have an action $S[A]$, a configuration is called a "classical solution" if it extremizes the action, or equivalently if it satisfies the equations of motion. In gauge theory on four-manifolds, you construct an honest moduli space of instantons $\mathcal{M}_{\text{inst}}$. There are anti-self dual connections $A$ which can be shown to minimize the Yang-Mills functional $S_{\text{YM}}[A]$. Therefore, I'd naively say that instantons are classical since they minimize an action. (BTW do they even solve the vacuum Yang-Mills equations?)
On the other hand, many beautiful results in both math and physics come by integrating over $\mathcal{M}_{\text{inst}}$ or similarly, computing SUSY invariants of $\mathcal{M}_{\text{inst}}$ like Euler characteristic, elliptic genus etc. Such a global topological invariant of a moduli space sounds pretty quantum to me: it sounds like a path integral with a particular choice of a measure. In addition, I hear people saying things like "instantons are suppressed" which makes them sounds like quantum corrections or something.
So what's the right way to think of all this? Should I think of $\mathcal{M}_{\text{inst}}$ as a "moduli space of classical vacua"? Then what does it mean to integrate over a moduli space of classical configurations vs. a moduli space of all configurations? For example in gauge theory, we write $\mathcal{A}/\mathcal{G}$ as the space of all connections modulo gauge transformations. What is the relation between
$$\int_{\mathcal{M}_{\text{inst}}} \cdots \,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\, \int_{\mathcal{A}/\mathcal{G}} \cdots \,\,\,\,\,\,?$$
