Suppose $Y \subset X$ is a closed subscheme of $X$. If $I$ is the ideal sheaf associated to $Y$, we have a short exact sequence
$$ 0 \rightarrow I \rightarrow \mathscr O_X \rightarrow \mathscr O_Y \rightarrow 0, $$
where $I = \mathscr O_X(-D)$, and $D = Y$ is an effective divisor.
My question is, given any effective divisor $D$, is there always such a sequence? What about effective divisors that are not irreducible, not reduced, or neither of the two? What about effective $\mathbb Q$-divisors?
Suppose for instance $D = 2F$, where $F$ is prime. We do have a sequence $$ 0 \rightarrow \mathscr O_X(F) \rightarrow \mathscr O_X(D) \rightarrow \mathscr O_F(D) \rightarrow 0. $$ If we take $$ 0 \rightarrow \mathscr O_X \rightarrow \mathscr O_X(D) \rightarrow Q \rightarrow 0, $$ what is $Q$, in other words what is $\mathscr O_X / \mathscr O_X(-D)$? Does it even make sense to speak of $D$ as a scheme, and therefore of $\mathscr O_D$?
Thank you!