$\DeclareMathOperator\rad{rad}$Question 1 is a special case of the following statement for finite dimensional algebras: Let $A$ and $B$ be a finite dimensional algebras over a field $K$ such that $A/\rad(A)$ and $B/\rad(B)$ are isomorphic to a direct product of matrix algebras over $K$ (which is always true when $K$ is algebraically closed).
When $e_i$ and $e_i'$ are pariwise orthogonal primitive idempotents which sum to 1 for $A$ and $B$ respectively, then $e_i \otimes_K e_i'$ are pairwise orthogonal primitive idempotents which sum to 1 for $A \otimes_K B$. Thus when $S_i$ and $S_i'$ are the simple $A$ and $B$-modules respectively, then $S_i \otimes S_i'$ are the simple $A \otimes_K B$ modules. This is proved for $A=B$ in the book "Frobenius algebras I" by Skowronski and Yamagata in chapter IV as proposition 11.3, but the proof works in exactly the same way when $A \neq B$.
