In the survey paper

https://arxiv.org/abs/1004.2583

of Bauer-Catanese-Pignatelli, they mention a question of Mumford:

Can a computer classify all surfaces of general type with $p_g=0$?

I've been playing a bit with the Craighero-Gattazzo (CG) surface (a particular surface of this kind) using various computer algebra systems, and my life has been made difficult by the fact that the standard equations for this surface are defined over a cubic extension of $\mathbb{Q}$ rather than over $\mathbb{Q}$ itself (it seems to make computations run for longer and various algorithms are not implemented).

This made me worry about Mumford's question: because $\bar{\mathbb{Q}}$ is only countable, a generic complex surface in this moduli space will only be defined over some transcendental extension of $\mathbb{Q}$, which presumably makes Groebner basis calculations even less tractable. So my question is:

In the moduli space of general type surfaces, is anything known about the existence or density of surfaces defined over $\mathbb{Q}$ or $\bar{\mathbb{Q}}$? Should I be able to perturb the pluricanonical ring of the CG surface and find a "nearby" surface defined over $\mathbb{Q}$? Should every component of moduli space contain a surface defined over $\bar{\mathbb{Q}}$?

If this question is too general, I would be happy to know the answer to the following more concrete question.

The CG surface has an explicit birational model as a quintic in $\mathbb{P}^3$ with four simple elliptic singularities. The standard model is defined over $\mathbb{Q}[r]/(r^3+r^2-1)$. Is it known that it is necessary to work over this cubic extension, or could there be a similar quintic defined over $\mathbb{Q}$ with the requisite properties, i.e. whose minimal resolution is biholomorphic to (or at least deformation equivalent to) the CG surface?