Joel, it seems to me that the sup of these metadefinable ordinals is exactly the first 1-stable. The key observation is that it's $\Sigma_1$ to say that a specific metadefinable ordinal is countable, and so there'll be an ordinal meta-defined by the ability to see this collapse map. This implies that the supremum $\sigma$ will have $L_\sigma$ ($\Sigma_1$-)pointwise definable. But then any $\Sigma_1$ formula true in $L$ with parameters from $L_\sigma$ is equivalent to a $\Sigma_1$ sentence. And so it must be true in some $L_\beta$ below $L_\sigma$ and hence in $L_\sigma$ by upward absoluteness.

Might be relevant: if you add a Sacks real $s$ over $L$, then in $L(s)$, for every real $x,y$, there is a continuous map coded in $L$ that maps $x$ to $y$. This shows up as Lemma 74 in On Sacks Forcing and the Sacks Property by Stefan Geschke and Sandra Quickert.

There's a chapter titled The Raisonnier Filter in Ralf Schindler's textbook (Set Theory: Exploring Independence and Truth). It presents the Lebesgue measure result in a learner-friendly manner, I think.

In fact, the hypothesis is not necessary for the existence of a largest $\Pi^1_1$ set without a perfect subset. Provably in ZF+DC, the set $\{x\in \mathbb{R} \mid x\in L_{\omega_1^x} \}$ is such a set.

(continued) From a cursory look at the proof in Jech. I think the proof of (2) carries over to the case of forcing with Silver collapse. But of course, I could be wrong.

Asaf, I think for some ground model and $U$, it's possible for there to be no generic filter for the Levy collapse in this situation. For example, start with $V_0=L[U]$ with a measurable $\kappa$ and normal measure $U$. Forcing to collapse it to $\omega_1$ with Silver collapse, then in $V=V_0[G]$, $\kappa$ is $\omega_1$ and has a precipitous ideal. If we happen to pick the $U$ we started with, then $L[U]$ here is just the ground model. But since $V$ is a Silver collapse extension, there is no $V_0$-generic filter for Levy collapse (see here).

@喻良 yes, that's right. Thanks for catching that typo!

With respect to the specific problem of requiring a specific closure, this is treated for instance in Jech by Lemma 21.9.

Since we'll need to disturb the continuum function very often below $\kappa$, and we won't know beforehand what the last bit of the forcing will look like (i.e. the poset named by the final factor in the iteration, to add $\kappa^{++}$ subsets to $\kappa$), so we let iteration take care of that by letting the final factor just be "that poset to add this many subsets to $\kappa$ when we get there".