Are all the theorems true?

$\def\zfc{\mathrm{ZFC}}\def\pa{\mathrm{PA}}$First, there is no consistent recursively axiomatizable theory extending Robinson’s arithmetic which has the property of having existential witnesses as described by Sridhar Ramesh. Let $\pi=\forall x\,\theta(x)$ be a true but $T$-unprovable $\Pi^0_1$ sentence with $\theta$ bounded, which exists by Gödel’s theorem. Then $\exists y\,(\theta(y)\to\forall x\,\theta(x))$ is a tautology, but there is no $n\in\omega$ such that $T\vdash\theta(n)\to\forall x\,\theta(x)$: since $\pi$ is true, $\theta(n)$ is provable in Robinson’s arithmetic, hence $T$ would prove $\pi$.

In fact, an iteration of the same idea shows that the only consistent theory with the property of having existential witnesses is the true arithmetic $\mathrm{Th}(\mathbb N)$.

The situation with goodness is more complicated: there are good theories, such as any consistent theory axiomatizable over $\pa$ by a set of $\Pi^0_1$ sentences. Nevertheless, neither $\zfc$ nor any its recursively axiomatized extension is good.

Let $T=\zfc$, or more generally, let $T$ be any recursively axiomatizable extension of $\pa$ which proves the local $\Sigma^0_1$-reflection principle for $\pa$. Let $\Box_\pa$ denote the provability predicate for $\pa$, and $T_{\Pi^0_1}$ the set of all $\Pi^0_1$ theorems of $T$. By a theorem of Lindström, there exists a $\Pi^0_1$ sentence $\pi$ such that $\pa+\pi$ is a $\Sigma^0_1$-conservative extension of $\pa+T_{\Pi^0_1}$. $T$ proves the reflection principle $$\tag{$*$}\Box_\pa(\neg\pi)\to\neg\pi$$ which can be written as a $\Sigma^0_2$ sentence, hence assuming $T$ is good, $(*)$ is provable in $\pa+T_{\Pi^0_1}$, and a fortiori in $\pa+\pi$. But then $\pa+\pi$ proves its own consistency, hence by Gödel’s theorem, it is inconsistent. By $\Sigma^0_1$-conservativity, $\pa+T_{\Pi^0_1}$ is also inconsistent, hence $T$ is inconsistent, contradicting its goodness.

Reference:

Per Lindström, On partially conservative sentences and interpretability, Proc. AMS 91 (1984), no. 3, pp. 436–443.

"goodness" appears to be your attempt to describe the property of having existential witnesses (that whenever $T$ proves there exists an $x$ such that $P(x)$, there is also a specific numeral n such that $T$ proves $P(n)$). There are also other related ideas such as $\omega$-consistency. But "goodness" doesn't quite match any of these exactly.

Re: question 3: No, "goodness" is not equivalent to consistency. After all, PA plays an unduly special role in the definition of "good", and not such a special role in the idea of "consistent". An example of a theory which is consistent (on the standard assumption that PA is) but not "good" is PA + "PA is inconsistent". This proves the $\Sigma_1$ statement "PA is inconsistent", but no $\Pi_0$ statement entailing this in PA.