Few Questions about Contraction of Exceptional Curve $E$ on a Smooth Surface

Let $X,Y$ be two algebraic surfaces (=smooth, proper $2$-dim schemes over fixed base field $k$) and let $E \subset X$ exceptional curve, ie $E \cong \Bbb P^1$ with self intersection $E^2=-1$. By Castelnuovo's contraction theorem $E$ can be contracted to a smooth point of a smooth surface leaving the rest of surface untouched; ie there exists a unique $q: X \to S$ with smooth $S$ such that $s:=q(E)$ a point and $X-E \cong S-q(E)$; this process is called usually "quadratic transformation" or blowdown (for details and properties, see Hartshorne's book).

Assume that there is a given a proper surjective morphism $f:X \to Y$ such that $f(E)= y$ is a point.

Few questions:

Question 1: How to see then $f$ must factor as morphism of schemes(!) through $q: X \to S$, ie there exist $r: S \to Y$ such that $f= r \circ q$?
Note that "set theoretically" or topologically that's clear, the question is why this factorization happens in category of schemes, ie $f= r \circ q$ with $r$ a morphism?

Let's now switch to relative situation, ie assume that $X,Y$ are relative schemes over some common base (smooth) curve $B$ (or maybe even any base scheme $B$), so we have data $x: X \to B, y: Y \to B$ and $f: X \to Y$ is a $B$-morphism.
Assume that the exceptional curve $E \subset X$ sits moreover in a fibre, ie there exists $b \in B$ such that $E \subset x^{-1}(b)$ (=scheme theoretic fibre).

Question 2: Does under these assumptions the "quadratic transformation" or blowdown $q: X \to S$ respect the base, ie is the blowdown $q:X \to S $ even a $B$-morphism?
And as natural follow up, is the factorization from first question $f= r \circ q$ behave well with base $B$, ie give a factorization $f= r \circ q$ of $B$-morphisms? (If yes, how to see it? If that's to general, assume that $x: X \to B, y:Y \to B$ are fibrations)

These two questions are motivated by proof of Theorem 10.21. (p 155) in Badescu's Algebraic Surfaces. There the author seems to take the first part for granted, but it seems to me that it is not clear why the factorization $f= r \circ q$ is "schematically" valid and if it is possible to extend this argument to relative setting which one contracts an exceptional curve sitting inside a fibre of a fibration.

Question 3: (Maybe I should pose this part separately?) Above the quadratic transformation/blowdown contracts curve $E$ if it is $\cong P^1$ and $E^2=-1$ to a point $s=q(E)$ of a smooth surface $S$. Say $E$ is a smooth irreducible curve on smooth surface $X$ with negative self intersection number $E^2=-d$ (Note, by adjunction formula, $0 \le d \le 2$).

When can $E$ be contracted to point of a now not neccessarily smooth algebraic surface $T$? Ie $h: X \to T$ such that $h(E)= t$ with $X-E \cong T-h(E)$? Can the completion of local ring $\mathcal{O}_{T,t}$ be described explicitly (maybe in terms of local equation of $E$)?
Which properties this singular surface $T$ has locally around $t$? Is it normal, Cohen-Macaulay, etc? How "bad" the singularity at $t$ could actually happen to be?