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 $A \neq B$.
For question 2, one can find in section 4.5. corollary 4.16 in the book "A tour of repreesntation theory" by Martin Lorenz the fact that $\mathbb{Q}$ is a splitting field for the symmetric group. The whole section 4 in this book is dedicated to the representation theory of the symmetric group in characteristic 0 and might be one of the nicest modern approaches to this problem.
Thus since $\mathbb{Q}$ is a splitting field for the symmetric group, it is true for any field $K$ of characteristic 0 (not just algebraically closed fields) that the irreducible representations of a direct product of symmetric groups is given as a tensor product of the irreducible representations of the single symmetric groups.
