I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. every algebra (in a countable language) of size $\kappa$ possesses a proper subalgebra also of size $\kappa$. V = L implies there are no Jonsson cardinals; $\aleph_0$ and many other cardinals are ZFC-provably not Jonsson cardinals; section 8 and some of section 23 of A. Kanamori, The Higher Infinite, contain a succinct account of the relevant results.

Suppose $F$ is a family of standard models of ZFC that extend $V$ and have the same ordinals. Let us say the pair $(V, F)$ is Jonsson-friendly if whenever $\kappa$ is Jonsson in $V$, then $\kappa$ is Jonsson in $W$ for every $W \in F$. A forcing $\mathbb{P}$ is Jonsson-preserving if $(V, V^{\mathbb{P}})$ is Jonsson-friendly.

By a result of Erdos and Hajnal, a sufficient condition for $(V, F)$ to be Jonsson-friendly is that $V$ has the following approximation property for every $W \in F$:

(JP) for every binary commutative function $f : \kappa \times \kappa \rightarrow \kappa$ in $W$, there exists $F \in V$ such that $V \models (F : \kappa \times \kappa \rightarrow [\kappa]^{\leq \omega}$ is a function$)$ and $(\forall \alpha, \beta < \kappa)(f(\alpha, \beta) \in F(\alpha, \beta))$.

It follows that for example every ccc forcing is Jonsson-preserving.

Q1: Can one characterize and describe the class of Jonsson-preserving forcings and any operations under which the class is closed?