Actually I just realized that complete intersections already give a superpolynomial number. Presumably one can get better bounds with a bit more cleverness.

@JasonStarr I was hoping you would see this question; do you know if this question has previously been considered?

@Mohan: I'm pretty sure $n=4$ is fine; IIRC, all smooth complete intersection surfaces of nonnegative Kodaira dimension satisfy $c_1^2\leq 2c_2$.

Perhaps I'm misunderstanding the question, but I'm fairly confident that $\mathfrak{S}(n)$ is not known to be finite.

A general type surface (in characteristic 0) cannot be swept out by a family of elliptic curves (this follows from a short argument with pluricanonical forms.) This implies that the number of elliptic curves on a general type surface is countable; a deep conjecture (closely related to Bombieri-Lang) predicts that there are only finitely many elliptic curves.

I suspect the issue is that the KodairaDimension function is only meant for mildly singular surfaces. The webpage magma.maths.usyd.edu.au/magma/handbook/text/1354 lists the related function KodairaEnriquesDimension as only working for surfaces with ADE singularities.

The answer to your first question is yes (you can see this e.g. by taking taking a basis $v_i$ of $V$ with $v_1,v_2,\cdots v_n$ a basis of $\pi^{-1}([\omega])$ and writing $\omega$ in terms of the induced basis of $\wedge^nV.$)

Not necessarily; every map from C to an abelian variety factors through the Jacobian, and there will be a map from the Jacobian of C to an elliptic curve iff the Jacobian of C is isogenous to a product of two elliptic curves. And every abelian surface is the Jacobian of a genus 2 curve, so just choose a curve corresponding to an abelian surface that isn't isogenous to the product of two elliptic curves.

An observation: If one exists, it must contain all irreducible degree 5 rational curves through those 8 points. Now if the 8 points can be degenerated so that this space is larger than usual, then you can probably prove nonexistence...