My motivation is this Wikipedia article.
Suppose $R$ be a ring with unity and $M,N$ be $R$-modules.
Take their direct sum/product $P=M \oplus N$. So $P$ is also a $R$-module. Consider the respective rings of endomorphisms $$E_1:=\text{End}_R(M),~ E_2:=\text{End}_R(N),~E_3:=\text{End}_R(P).$$
What is the relation among $E_1,E_2,E_3$ ? Is $E_3=E_1 \oplus E_2$ ?
Is the story same for direct product of two algebraic object e.g., for two rings $R,S$, does $\text{End}(R \times S)=\text{End}(R) \times \text{End}(S)$ ?
Here is some information about homomorphisms between direct sums. Let $\mathcal{C}$ be a abelian category of algebraic objects $\{A_i\}_{i \in I}$ with a finite index set $I$. We can equip the direct sum $\oplus A_i$ with projection homomorphism $\pi_j: \oplus A_i \to A_j$ for each $j \in I$ as well as we can equip with coprojection $\alpha_j:A_j \to \oplus A_i$ for each $j \in I$.
Does this help to answer my question 1 and 2 ?
$\DeclareMathOperator{\End}{End}\DeclareMathOperator{\Hom}{Hom}$ In general, it is not true that $E_3=E_1 \oplus E_2$. Perhaps the simplest example is that of a vector space $P$ of dimension $2$ over a field $R$.
Addendum
$E_{3}$ can be readily described in terms of matrices $$ \begin{bmatrix} \mu & \alpha\\ \beta & \nu \end{bmatrix}, $$ where $\mu \in \End_{R}(M)$, $\nu \in \End_{R}(N)$, $\alpha \in \Hom(N, M)$, $\beta \in \Hom(M, N)$, acting naturally $$ \begin{bmatrix} \mu & \alpha\\ \beta & \nu \end{bmatrix} \begin{bmatrix} m\\ n \end{bmatrix} = \begin{bmatrix} \mu(m) + \alpha(n) \\ \beta(m) + \nu(n) \end{bmatrix} , $$ where $m \in M$ and $n \in N$.
So for your relation to hold, you need $\Hom(N, M) = \Hom(M, N) = 0$. For instance, this is the case when $R = \mathbb{Z}$, and $M, N$ are finite abelian groups of coprime order.