These definitions are all from Hartshorne's Algebraic Geometry.

Let $X$ be a scheme.  For each open set $U$, let $S(U)$ be the subset of $O_X(U)$ such that $s \in S(U)$ if for every point $x \in U$, the stalk of $s$ at $x$ is not a zero divisor in $O_{X, x}$.  Then, $S(U)$ is a multiplicatively closed subset of $O_X(U)$, and the localization $U \rightarrow S(U)^{-1} O_X(U)$ forms a presheaf of commutative rings. Let $M$ be the sheafification of this presheaf.

The next step is to take $M^* / O_X^*$.  where $M^*$ and $O_X^*$ are the sheaves of groups of units of $M$ and $O_X$ respectively.  This is a sheaf of abelian groups.  Define a Cartier divisor to be a global section of $M^* /  O_X^*$, and define a principal Cartier divisor to be a Cartier divisor in the image of the canonical morphism $\Gamma(X, M^*) \rightarrow \Gamma(X, M^* / O_X^*)$. 

My questions are:

1. Why is $U \rightarrow S(U)^{-1} O_X(U)$ a presheaf of commutative rings? I'm having trouble defining how the restriction morphisms works.  If $V \subseteq U$, how do we define $S(U)^{-1} O_X(U) \rightarrow S(V)^{-1} O_X(V)$?

2. If we take $M^* / O_X^*$, then we'd expect there to be some kind of inclusion $O_X \rightarrow M$. I understand there is an inclusion morphism $O_X \rightarrow (U \rightarrow S(U)^{-1} O_X(U))$ , but why is it still injective after taking the sheafification map?

3. Why is $S(U)$ defined that way? What's wrong if we define $S(U)$ to be the set of non zero divisors of $O_X(U)$? or are they equivalent?