@AndreasBlass right, that was a silly reading I made. Thanks

I'm being ignorant here: what does "sufficiently generic" mean?

while this isn't quite what you're looking for, you might be interested to take a look at the appendix of Large cardinals need not be large in HOD and the references there to the works of Friedman and Brooke-Taylor on coding into the diamond pattern.

Done. I also like this line of work very much. There's also an earlier and shorter draft in 2016 before it was split into a two-parter.

Consistently the answer to question 1 is "exactly the inaccessibles". This is Theorem 8.5 in Inner Models from Extended Logics (Part 1), 2020: assuming $V=L$, then the models of ZFC(SOL) are exactly those isomorphic to models of ZFC of the form $L_\kappa$ where $\kappa$ is inaccessible.

@AsafKaragila I vaguely recall reading somewhere that this is an intentional attribution, but I don't remember where

An observation: if the mantle has the form $L[A]$, where $A$ is a class of ordinals, then we may define an artificial $A$-recovering quantifier $Q^A$ as follows: $N\vDash (Q^A xy)\varphi(x,y,\vec{a})$ iff $\{(x,y)\in N^2\mid N\vDash \varphi(x,y,\vec{a})\}$ is a linear order of ordertype in $A$. Then the resulting model is just $L[A]$.