PFA(c) has a $\Sigma^2_1$-indescribable cardinal as an upper bound in consistency strength (this is in Hierarchies of Forcing Axioms I. by Neeman and Schimmerling).

@NoahSchweber No problem. That's correct.

Theorem 3.1.12 of Larson's stationary tower book or theorem 7.22 of Steel's Outline of Inner Model Theory. Larson's book and Steel's exposition on the derived model theorem contain many (but not all) of the famous generic absoluteness results.

Here are some partial answers : Lemma 3.6 and 3.8 of Jackson's handbook chapter tell us that if the Steel pointclass beyond $\Gamma$ is inductive-like then it must be that $o(\Gamma)$ is regular (in our starting $\mathsf{AD}^+$ model). Any model of $\mathsf{LSA}$ shows us that it's real possibility that we have an inductive-like pointclass immediately after a pointclass satisfying $\mathsf{AD_{\mathbb{R}}+DC}$.

Woodin has a proof of this under the additional assumption that every set admits an $\infty$-Borel code. I don't believe it's known if it follows from just ZF + AD + DC.