Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$ It is known that in the extension there are no special $\aleph_2$-Aronszajn trees but there are $\aleph_2$-Aronszajn trees.

Question 1.Is there any $\aleph_2$-Souslin tree in $L[G]?$

I assume the believe is that there are, but I don't know how to prove it. Surprisingly, if, instead of $\aleph_2$, we consider a cardinal $\lambda^+ > \beth_\omega,$ with $\lambda$ regular and apply the Mitchell forcing to get $2^\lambda=\lambda^{++}=\kappa,$ then the results of Assaf Rinot show that there are $\lambda^{++}$-souslin trees in the extension.

My second question is motivated by the work of Laver-Shelah. Assume $V=L$ and $\kappa$ is weakly compact. In order to produce a model of $CH+$there are no $\aleph_2$-Souslin trees, Laver and Shelah, first force with Levy collapse $Col(\aleph_1, < \kappa),$ and over it do an iteration to kill all possible $\aleph_2$-Souslin trees.

Question 2.Are there any $\aleph_2$-Souslin trees just after doing the Levy collapse $Col(\aleph_1, < \kappa)$?