This is a clarificatory question regarding Asaf's answer to this question:
What axioms need to be added to second-order ZFC before it has a unique model (up to isomorphism)?
He describes the technique of defining a categorical extension of ZFC2 by adding an axiom asserting "There are exactly $\kappa$-many inaccessibles", where the extension is satisfied by a model of ZFC2 of the form $V_{\kappa_1}$. (Where $\kappa$ is a cardinal and $\kappa_1$ is the next inaccessible after $\kappa$)
(a) How many categorical extensions is it possible to obtain by this technique?
(b) If we wanted to describe proper-class-many categorical extensions, and we allow ourselves proper-class-many ordinals as parameters, would that make the technique described above work for proper-class-many inaccessibles? (I.e. if we use inaccessible ordinals in place of $\kappa$ in "$\kappa$-many"...)
(c) What sort of problems is he referring to if K $\cap V_{\kappa}$ is "really complicated" or $\kappa$ is really large, or if "crazy reflections" occur?
In general, there are countably many formulas in the language of set theory. Even the second-order language. So there are at most countably many extensions.
To your second question, if we allow ordinal parameters, then we can always add "There are exactly $\alpha$ inaccessible cardinals". This will characterize your model completely. But the problem is when $\alpha$ is internally the height of your model. This would cause a problem, since that axiom would be "there is a proper class of inaccessible cardinals". To overcome this, we can assume that there a proper class of inaccessible cardinals, but no inaccessible limit of inaccessible cardinals.
The third question is the inherent problem I mention above. If $\kappa$ is very large, then it is possible that any second-order properties of $\kappa$ are reflected to a proper class below it. And so there is no way to internally describe $V_\kappa$ in a unique way. So we cannot characterize such models internally with a single (or a schema of) second-order axioms. For example measurable cardinals have this property.