For the first problem, as I wrote in the comments, the author is implicitly using the language of $(\infty,1)$-categories, where it does make sense to speak of such a functor.
To understand why, you can consult any of the many introductions to $(\infty,1)$-category theory.

The rest of this answer deals with the second problem.
$\newcommand{colim}{\mathop{\mathrm{colim}}}
\newcommand{hocolim}{\mathop{\mathrm{hocolim}}}$
A fundamental property of any topos is universality of colimits, which means that colimits commute with base change:
$$ (\colim_i X_i) \times_B A \simeq \colim_i (X_i \times_B A) $$
for any morphism $A \to B$.
What Rezk calls the "distributive law" (Proposition 3.7) in his paper "Fibrations and homotopy colimits..." is just the special case of universality of colimits where $B = \colim_i X_i$.

In any "homotopy topos" or $\infty$-topos, there is the analogous property of commutativity of homotopy colimits with homotopy base change, i.e.
$$ (\mathrm{hocolim}_i X_i) \mathop{\times}^h_B A \simeq \mathrm{hocolim}_i (X_i \mathop{\times}^h_B A) $$
for any morphism $A \to B$.
In particular this holds in the archetypal example of a homotopy topos: the homotopy theory of simplicial sheaves.
As a special case one has a "homotopy distributive law".
Then Rezk's Theorem 1.4(1) in "Fibrations..." is a direct application of this homotopy distributive law.
Indeed, in the notation of loc. cit., one sees
$$\begin{align}
\hocolim_i X_i &\simeq \hocolim_i (\colim X \mathop{\times}^h_{\colim Y} Y_i)\\
&\simeq \hocolim_i (\colim X \mathop{\times}^h_{\hocolim Y} Y_i) \\
&\simeq \mathrm{colim} X
\end{align}$$
when $\colim Y \simeq \hocolim Y$.
For a reference for this in the language of model categories, see paragraph 6.5 in these notes of Rezk (this is the property P1).
In fact, the proof of the homotopy distributive law is the same as the proof of the ordinary distributive law as described by Rezk, using as a starting point the claim for spaces instead of for sets.

Since your question was rather about the proof in Rezk's older paper, let me try to say something about that.
Let me emphasize that I have not really read the paper, so I may say something wrong in what follows.
As far as I understood, the idea of the argument there is to use the distributive law for ordinary toposes, plus the explicit description of homotopy colimits of simplicial sheaves due to Bousfield-Kan.

For clarity, the proof can be divided into two steps: first, one demonstrates the claim assuming that $\colim X \to \colim Y$ is a sharp map.
Then in general one reduces to the above case by factoring this map as a weak equivalence followed by a sharp map.
The sharp version is:

**Lemma:**
Let $Y : I \to s\mathscr{E}$ be a diagram and let $p : W \to \colim Y$ be a sharp map in $s\mathscr{E}$.
Let $X : I \to s\mathscr{E}$ be the diagram defined by $X_i = Y_i \times_{\colim Y} W$ for each $i$.
If $\hocolim Y \to \colim Y$ is a weak equivalence, then $\hocolim X \to \colim X$ is a weak equivalence.

Proof:
First of all, the distributive law says there is an isomorphism
$$ \colim_i X_i \approx \colim_i (Y_i \times_{\colim Y} W) \approx W. $$
Using the assumptions that $p$ is sharp and $Y$ is a homotopy colimit diagram, one gets the chain of weak equivalences
$$\begin{align}
\colim X &\simeq \colim X \mathop{\times}^h_{\colim Y} \hocolim Y \\
&\simeq \colim X \mathop{\times}_{\colim Y} \hocolim Y \\
&\simeq \colim X \mathop{\times}_{\colim Y} \colim \tilde{Y} \\
&\simeq \colim_i (\colim X \mathop{\times}_{\colim Y} \tilde{Y}_i) \\
&\simeq \colim_i \tilde{X}_i \\
&\simeq \hocolim X
\end{align}$$
using at the end $\colim X \mathop{\times}_{\colim Y} \tilde{Y}_i \approx \tilde{X}_i$ (this is where one needs to open up the Bousfield-Kan constructions $\tilde{X}$ and $\tilde{Y}$, I believe, making use of the isomorphism $X_i \approx Y_i \times_{\colim Y} \colim X$ from above).