Existence of homotopy limits and colimits in model categories

Let me start first answering your second question (this is quite close to the idea of John Klein).

Definition. A pair $(\mathbf C,\mathcal W)$, with $\mathbf C$ a category and $\mathcal W$ a class of morphisms in $\mathbf C$, is said to be a category with weak equivalences provided:

1. $\mathcal W$ contains the class of all isomorphisms in $\mathbf C$;

2. $\mathcal W$ has the two out of three property.

Of course, for any category $\mathbf C$, the pair $(\mathbf C,\{\text{isos in $\mathbf C$}\})$ is a trivial example of category with weak equivalences. Roughly speaking, one may think to a category with weak equivalences as a category with a distinguished class of morphisms, that are not necessarily isomorphisms, but that "we want to consider as they were isomorphisms''.

Given a category with weak equivalences $(\mathbf C,\mathcal W)$, one would like to find a new category $\mathbf D$ obtained from $\mathbf C$ in a "universal way'', and where all the morphisms in $\mathcal W$ "become isomorphisms''. In the next definition we make this idea precise:

Definition. Given a category with weak equivalences $(\mathbf C,\mathcal W)$ we say that:

1. a functor $F\colon \mathbf C\to \mathbf D$ is $\mathcal W$-inverting provided $F(w)$ is an isomorphism in $\mathbf D$ for all $w\in\mathcal W$;
2. a universal localization of $(\mathbf C,\mathcal W)$ is a pair $(\mathbf C[\mathcal W^{-1}], Q\colon \mathbf C\to \mathbf C[\mathcal W^{-1}])$ of a category $\mathbf C[\mathcal W^{-1}]$ and a $\mathcal W$-inverting functor $Q$ such that, for any $\mathcal W$-inverting functor $F\colon \mathbf C\to \mathbf D$ there exists a unique functor $\bar F\colon\mathbf C[\mathcal W^{-1}]\to \mathbf D$ such that the following diagram commutes

Given a category with weak equivalences $(\mathbf C,\mathcal W)$, there is a standard construction of a localization $Q\colon \mathbf C\to \mathbf C[\mathcal W^{-1}]$ (see Gabriel-Zismann) but, in general, this construction may produce a category where some of the $\hom$-sets are proper classes. We say that a universal localization exists if and only if this standard construction produces a locally small category.

Consider two categories with weak equivalences $(\mathbf C,\mathcal W)$ and $(\mathbf C',\mathcal W')$ and suppose that both admit a universal localization, that we denote by $Q\colon \mathbf C\to \mathbf C[\mathcal W^{-1}]$ and $Q'\colon \mathbf C'\to \mathbf C'[(\mathcal W')^{-1}]$, respectively. Suppose also that we have a functor $F\colon \mathbf C\to \mathbf C'$ such that $F(\mathcal W)\subseteq \mathcal W'$. Then, the composition $Q'\circ F$ is $\mathcal W$-inverting and, by the universal property of the localization, there exists a unique functor $\bar F\colon \mathbf C[\mathcal W^{-1}]\to \mathbf C'[(\mathcal W')^{-1}]$ that makes the following diagram commute:

In the notation of the above discussion, for a general functor $F$, that is, when $F(\mathcal W)$ is not necessarily contained in $\mathcal W'$, there does not in general exist a functor $\bar F$ that makes the diagram strictly commutative. On the other hand, we still want to be able to define a functor on the localizations that is induced by $F$ in some universal way:

Definition. Given two categories with weak equivalences $(\mathbf C,\mathcal W)$ and $(\mathbf C',\mathcal W')$ whose universal localizations exist, and are denoted by $Q\colon \mathbf C\to \mathbf C[\mathcal W^{-1}]$ and $Q'\colon \mathbf C'\to \mathbf C'[(\mathcal W')^{-1}]$, respectively. Given a functor $F\colon \mathbf C\to \mathbf C'$, a left derived functor of $F$ is a pair $(\mathbf L F,\alpha)$ of a functor $\mathbf L F\colon \mathbf C[\mathcal W^{-1}]\to \mathbf C'[(\mathcal W')^{-1}]$ and a natural transformation $\alpha\colon \mathbf L F Q\Rightarrow Q' F$, which is universal in the following sense: given any other pair $(G,\beta)$ with $G\colon \mathbf C[\mathcal W^{-1}]\to \mathbf C'[(\mathcal W')^{-1}]$ and $\beta\colon G Q\Rightarrow Q' F$, there exists a unique natural transformation $\gamma\colon G\Rightarrow \mathbf L F$ such that $\alpha\gamma=\beta$.

This should answer your second question as follows:

Consider a model category $(\mathbf M, \mathcal W,\mathcal C,\mathcal F)$, so that, for a small category $I$, both $(\mathbf M, \mathcal W)$ and $(\mathbf M^{I},\mathcal W_I)$ (where $\mathcal W_I$ if the class of morphisms that are pointwise in $\mathcal W$) are categories with weak equivalences. One can consider the I-colimit functor $\mathbf M^I\to \mathbf M$ and the homotopy colimit functor can be defined as its left derived functor in the sense above. In this way you get a universal property.

As for your first question, there are various constructions of homotopy co/limits in the literature. The one I prefer is due to Chacholski and Scherer. They start from the following observation: in a model category $(\mathbf M,\mathcal W,\mathcal C,\mathcal F)$, the important part is the category with weak equivalences $(\mathbf M,\mathcal W)$ which allows to define (though not to compute) the homotopy category and derived functors, while the classes of fibrations and cofibrations, the "model structure", are just instrumental to compute the objects that $(\mathbf M,\mathcal W)$ already uniquely identifies.

Based on this observation they define "external" model structures, that they call model approximations, for a given category with weak equivalences. More precisely, given a category with weak equivalences $(\mathbf M,\mathcal W)$ a left model approximation is given by a model category $(\mathbf M',\mathcal W',\mathcal C,\mathcal F)$ and a pair of adjoint functors $l:\mathbf M'\rightleftarrows \mathbf M:r$ that satisfy a set of axioms, designed to allow one to use the model structure on $\mathbf M'$ "as if it was a model structure on $\mathbf M$".

The mere existence of a left model approximation for a category with waek equivalences $(\mathbf M,\mathcal W)$ is enough to conclude the existence of the (locally small) universal localization $\mathbf M[\mathcal W^{-1}]$ that, in analogy with model categories, one calls homotopy category, see Proposition 5.5 in the paper.

Now, for derived functors the thing is a bit more subtle: given two categories with weak equivalences $(\mathbf M,\mathcal W)$ and $(\mathbf N,\mathcal X)$, and a functor $F: \mathbf M\to \mathbf N$, a left model approximation $l : \mathbf M' \rightleftarrows \mathbf M : r$ is good for $F$ if the composite $F l : \mathbf M' \to \mathbf N$ sends weak equivalences between cofibrant objects to weak equivalences.

The existence of a left model approximation which is good for a given functor allows one to give quite an explicit construction of its left derived functor, see Proposition 5.9 in the paper.

Now, let $(\mathbf M,\mathcal W)$ be a category with weak equivalences, let $I$ be a small category and suppose that $I$-colimits in $\mathbf M$ exist, so that we can consider the colimit functor $\mathrm{colim}_I:\mathbf M^I\to \mathbf M$. Suppose also that $(\mathbf M,\mathcal W)$ has a left model approximation. In Chapter 2 of their paper, Chacholski and Scherer, give an explicit construction of a model approximation for the category with weak equivalences $(\mathbf M^I,\mathcal W_I)$ which is good for the $I$-colimit functor. Hence, one can left derive that functor, obtaining the homotopy colimit functor.

Let me remark that they also describe suitable model approximations for general homotopy Kan extensions.

The answer to your first question should now be the following:

Given a model category $(\mathbf M, \mathcal W,\mathcal C,\mathcal F)$ there are a trivial left model approximation

$id:(\mathbf M, \mathcal W,\mathcal C,\mathcal F)\rightleftarrows (\mathbf M, \mathcal W):id$

and a trivial right model approximation

$id:(\mathbf M, \mathcal W)\rightleftarrows(\mathbf M, \mathcal W,\mathcal C,\mathcal F):id$

so the above results apply to give good left model approximations for all colimit functors and good right model approximations for all limit functors. In particular, homotopy co/limits exist in any model category.

This is mostly a comment to Simone's thorough answer. It addresses a different, arguably more natural, way to deal with categories of functor $\mathbf{M}^I$, with $\mathbf{M}$ a model category; but I have nothing to add with respect to Ivan's questions.

In Catégories dérivables, Cisinski shows in Proposition 6.21 that the prederivator associated to a left derivable category (1.1) which is homotopy complete (6.1) is in fact a left derivator with domain the 2-category of small categories. The same fact is generalised to $\infty$-categories in §7.7 of Cisinski's book Higher Categories and Homotopical Algebra, although with a slightly different emphasis and terminology, where left derivable $\infty$-categories are called $\infty$-categories with weak equivalences and fibrations.

This implies in particular that the homotopy limit functor $\text{holim} \colon ho(\mathbf{M}^I) \to ho(\mathbf{M})$ exists for any small category $I$. Notice that any Quillen model category is homotopy complete as a category with weak equivalences and fibrations, by standard closure arguments. The nice technical side of this theory amounts to the fact that the category $\mathbf{M}^I$ can be endowed with a canonical structure of category with weak equivalences and fibrations, which are defined component-wise (this is Theorem 6.11 of Catégories dérivables and was proved by Rădulescu-Banu in his thesis).