Reason for the differences in third ionisation energy between elements (magnesium and fluorine)
The ionisation energy, as a rough guide, can be related to the effective nuclear charge and the principal quantum number:

$$\mathrm{IE} \propto \left(\frac{Z_\mathrm{eff}}{n}\right)^2$$

In the ionisation of $\ce{Mg^2+}$ and $\ce{F^2+}$, both electrons are being ionised from the second principal quantum shell, so $n = 2$ in both cases. However, the main reason why the 3rd IE of magnesium is larger is because the nuclear charge $Z$ is larger, and not because of any so-called stability of half-filled or fully filled shells.

Contrary to what you thought, the nuclear charge is not the charge on the species. It is the charge of the nucleus - hence "nuclear charge". The $\ce{Mg}$ nucleus has 12 protons, and the $\ce{F}$ nucleus has 9 protons. So, there is a greater electrostatic force pulling the electron towards the nucleus in $\ce{Mg^2+}$ than in $\ce{F^2+}$, and consequently it is harder to remove the electron from $\ce{Mg^2+}$.

In truth, the raw nuclear charge must be modified by the shielding constant $\sigma$ to obtain the effective nuclear charge, $Z_\mathrm{eff}$. However, the variation in $\sigma$ between the two compounds is really negligible compared to the variation in the actual nuclear charge.