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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$Y=pt$. 1 The answer to your first question is yes. For$E_i$as above, the sequence $$0\rightarrow E_1\otimes\mathcal{O}_X(L) \rightarrow E_2\otimes\mathcal{O}_X(L) \rightarrow E_3\otimes\mathcal{O}_X(L) \rightarrow 0$$ is certainly also exact since$L$is flat as an$O_X$-module (since it is locally free). For the second question, if$S\$ is a non-flat sheaf, the sequence is in general not exact. I'll leave it to you to cook up some counterexample.