The answer to your first question is yes. For $E_i$ as above
Yes, the sequence $$0\rightarrow 0\rightarrow E_1\otimes\mathcal{O}_X(L) \rightarrow E_2\otimes\mathcal{O}_X(L) \rightarrow E_3\otimes\mathcal{O}_X(L) \rightarrow 0$$ 0\,\,$ is certainly also exact since $L$ is flat as an $O_X$-module (since it is locally free)free (hence flat).
For the second question, if $S$ is a non-flat sheaf, the sequence answer is negative in generalnot exact. I'll leave it Take $0 \to you I_Y \to cook up some counterexample.O_X \to O_Y \to 0$ and $S=O_Y$, where $X=\mathbb{A}^1$ and $Y=pt$.
