If $M\bigoplus N $ submodule $A\bigoplus B$ does it imply either $M$ submodule $A$ or $M$ submodule $ B$? 
If $M\bigoplus N$ is a submodule of $A\bigoplus B$ does it necessarily  imply either $M$ is a submodule of $A$ or $M$ is a submodule of $B$?

If in general it is not true, is it true if $M$ and $N$ are $\mathbb Z$-modules?
I tried finding counter examples when they are $\mathbb Z$-modules but did not succeed. 

No,
Let $A,B = \mathbb{Z}$ as $\mathbb{Z}$-modules.  Let $M$ be the submodule of $A\oplus B$ generated by $(2,1)$ and let $N$ be the submodule generated by $(0,1).$  Then $M\oplus N \subset A\oplus B$ but $M$ is not a subset of $A$ or $B.$ 
