I understand your theory to be the set of all sentences in the first-order language of set theory that are true in every model of $\newcommand{\ZFC}{\text{ZFC}}\ZFC_2$. This is a perfectly sensible notion that we can define even when working in $\ZFC$, which provides its notion of how to interpret the second-order theory $\ZFC_2$.
Namely, Zermelo famously proved that the models of $\ZFC_2$ are precisely those isomorphic to the transitive sets of the form $V_\kappa$, where $\kappa$ is an inaccessible cardinal. The nature of $\ZFC_2$, therefore, depends on whether there are any inaccessible cardinals and indeed how many there are. For example, if there is only one inaccessible cardinal, then $\ZFC_2$ is a categorical theory, but not otherwise, and Zermelo's result is often described as a quasi-categoricity result, since it shows that for any two models of $\ZFC_2$, one of them is an isomorphic to an initial segment of the other.
So your theory is simply
$$T=\bigcap\left\{\ \strut\text{Th}(\langle V_\kappa,{\in}\rangle) \mid \kappa\text{ is inaccessible }\right\}.$$
The exact nature of this theory will depend, however, on whether there are any inaccessible cardinals or indeed how many there are, as well as on other set-theoretic assertions in the context in which we define the theory.
If there are no inaccessible cardinals, for example, then $\ZFC_2$ has no models and so your theory has all sentences in it, an inconsistent theory. So it is possible that your theory is Very Strong.
If there is an inaccessible cardinal, then your theory includes all the arithmetic consequences of an inaccessible cardinal, which includes $\newcommand{\Con}{\text{Con}}\Con(\ZFC)$ and $\Con(\ZFC+\Con(\ZFC))$ and so forth, iterated many times. So the answer to question 1 is yes.
If there is an inaccessible cardinal, then your theory does not include the statement "there is an inaccessible cardinal", since indeed this statement is not true in $V_\kappa$ when $\kappa$ is the least inaccessible.
If there is exactly one inaccessible cardinal, then your theory is the complete theory of $V_\kappa$ for this unique inaccessible cardinal $\kappa$. (This could be seen as an analogue of what you had mentioned with the reals, where we got a complete theory, the theory of real-closed fields.)
If there is more than one inaccessible cardinal, then your theory is not complete, forming only part of the theory of $V_\kappa$ where $\kappa$ is the least inaccessible. For example, the assertion "there is an inaccessible cardinal" will be independent of your theory, since it is true in some $V_\kappa$ but not all. (Although the theory is not complete, the extra inaccessible cardinals have strong consequences down low, such as additional consistency assertions, which will be a part of $T$.)
If there are infinitely many inaccessible cardinals, then the assertion "there are $n$ inaccessible cardinals" is independent of your theory, since it is true in some $V_\kappa$ and false in others.
Similarly with "there are $\alpha$ many inaccessible cardinals", if there are more than $\alpha$ many inaccessible cardinals.
The theory $\ZFC+$ there is an inaccessible cardinal proves the consistency of your theory, since your theory is true in $V_\kappa$ for any inaccessible cardinal $\kappa$.
If V=L holds, then this will be part of your theory, since all the $V_\kappa$ for $\kappa$ inaccessible will think that it is true.
Similarly, as Noah mentions in the comments, if CH is true, then it is part of your theory, and similarly with $\neg$CH, or $\Diamond$, or SH, or MA, and so forth.
There is sometimes a confusion that arises when considering a second-order theory such as $\ZFC_2$. If we work as usual in the background theory of (first-order) $\ZFC$, then as I mentioned we can interpret the second-order theory of $\ZFC_2$ as it is defined in set theory. In this process, one views Zermelo's theorem, for example, as a theorem of $\ZFC$.
But sometimes people have the idea that there is a "true" second-order logic, not merely the semantics as it is defined in first-order set theory, and this is not the same, since for example we can have nonstandard models of $\ZFC$, and they would provide only an internal meaning of $\ZFC_2$, which would not accord with the "true" second-order semantics. The question whether there is such a meaningful semantics amounts to staking a claim for monism in the raging debate on pluralism in the philosophy of set theory.
You might be interested in the interplay of your theory with the analysis that Robin Solberg and I provide in our paper on categorical cardinals, which involves a similar take on looking at the first and second-order theories of $V_\kappa$ for $\kappa$ inaccessible. The paper is:
Meanwhile, let me conclude with the following scheme.
Theorem. If Ord is definably Mahlo (every definable class club contains an inaccessible cardinal) and $\sigma$ is true, then $\neg\sigma\notin T$.
This can be formalized as a scheme, and there is a level-by-level version where we assume only suitable $\Sigma_n$-Mahloness, above the complexity of $\sigma$.
Proof. If Ord is definably Mahlo and $\sigma$ holds, then by reflection $\sigma$ holds on a definable class club and therefore in some inaccessible $V_\kappa$. So $\neg\sigma$ is not in $T$. $\Box$
