I am confusing with the following observation: Let $f : X:=\mathrm{Spec}(K) \rightarrow \mathrm{Spec}(k) =: Y$ be a scheme morphism corresponding to a non-trivial finite field extension ( hence $f$ is proper and $X$, $Y$ are noetherian). Its Stein factorization is $ X \overset { \mathrm{id}_X} \longrightarrow X \overset {f} \longrightarrow Y$. But it's clear that $f$ is not a birational, even if $char=0$. We don't have

$f_{*} \mathcal{O}_X = \mathcal{O}_Y$

in this example ( $char = 0 $ or not) either, although we have $f$ has connected fiber and $Y$ is normal.

When I read the following discussion, I got a question.

When will the pushforward of a structure sheaf still be a structure sheaf?

Let $f: X \rightarrow Y$ be a proper morphism of noetherian shcemes and $ X \overset {f'} \rightarrow Z \overset {g} \rightarrow Y$ be its Stein factorization. In J.C. Ottem response, he mentioned that if the fibers of $f$ are connected, then $g$ must be birational ( I think that we assume both $X$ and $Y$ are integral schemes), and from this, one gets $g_{*} \mathcal{O}_Z = \mathcal{O}_Y$, hence we have

$f_{*} \mathcal{O}_X = \mathcal{O}_Y$

He also give the reference, which is Hartshorne III.10.3, for which I think the right one is Hartshorne III.11.3, but 11.3 is telling that the isomorphism gives connected fibers.

So I would like to know :

(1) How to see that $g$ is birational? The correct reference?

(2) In which part we need the characteristic 0 condition?

(3) Could we replace the characteristic 0 condition to another condition, e.g $f$ has integral fibers?