There are set theoretic issues that will hinder any proof of this statement. In particular I don't believe you can construct the adjunction in part (2) without applying the axiom of choice to CAT, and it's not clear the localization S^(-1)CAT makes sense either.
That said, let's ignore these issues and pretend that CAT is a small category of (small) categories.
I was originally confused by this. I believe that there are some problems in the formulation of this question. In particular if S consists of those functors which are equivalences of categories, then it does not form a multiplicative system. Specifically, the right Ore condition fails. I'll give an example later. What this means though, is that the property your are trying to check in part (1) actually fails. To see this pick a group G and view it as a category with one object. Then the functors from H to H are the homomorphisms, and the equivalence classes of this functors are the orbits of Hom(H,H) under the conjugation action. Choose H such that there are distinct automorphisms of H which are equivalent under conjugation.
Claim: If F,G: H --> H are two such automorphisms, then there is no equivalence s:H --> C such that sF = sG. I think this is easy to check, so I'll leave it at that.
So the second part of (1) is impossible to prove.
I first thought that we can answer this question be going to skeletal categories. This is not correct, but let's look at why it is not correct. Let SKEL be the full subcategory of CAT consisting of the Skeletal categories, i.e. those categories in which $a \cong b$ implies $a = b$, i.e. those categories which have a single object in each isomorphism class.
Every object in CAT is equivalent to one in SKEL. This is an easy exercise you should work out for yourself. By applying the axiom of choice to all of CAT we can construct an equivalence between CAT and SKEL, and in particular we have L: CAT ---> SKEL with an equivalence $C \cong L(C)$ for every category C.
Notice that a functor between skeletal categories is an equivalence if and only if it is an isomorphism on the nose. So we are part way there. This made me think that SKEL was the localizing subcategory we were after.
However two functors into a skeletal category can be naturally isomorphic without being equal. The group example above is actually an example of this. The functors F and G are naturally isomorphic, but not equal.
Let J denote the "Joyal interval", the category with two objects and an isomorphism between them. J can be used to say when two functors are naturally isomorphic. F,G: C --> D are nat. isomorphic if they extend to a functor $C \times J \to D$. There is a quotient of J where we identify the two objects. This is in fact the category Z (the group of integers viewed as a category). Let W be the set of morphisms consisting of the single morphism J --> Z. Notice that SKEL consists of exactly the "W-local" objects. But this is not quite what we want.
Okay, now for the rest. First, I promised an example where the right Ore condition fails. This is given by considering the two inclusions of pt into J. Both of these are equivalences, but this cannot be completed to an appropriate square. So the class of equivalences in not a multiplicative system.
Next, Toen doesn't assume that S is a multiplicative system. He (correctly) defines the localization in terms of a (weak) universal property. Namely, there exists a locization functor:
$ L: CAT \to S^{-1}CAT$
such that fro any other category D, $L^* : Fun( S^{-1}CAT, D) \to Fun(CAT, D)$ is fully-faithfull and the essential image consists of those functors which send elements of S to isomorphisms in D. In particular the quotient functor from CAT to |CAT| does this, so this gives us functor from S^(-1)CAT to |CAT| defined up to unique natural isomorphism.
On the other hand, |CAT| is also defined by a universal property. This gives us maps between |CAT| and S^(-1)CAT and by the usual sort of general nonsense this is an equivalence. The details aren't too hard, so I'll leave them to you.
Now the second part of your question asks about whether we can realize |CAT| as a full subcategory of CAT such that the quotient $ q:CAT \to |CAT|$ is left adjoint to the inclusion. The answer is no. Let's first look at whether q can have a right adjoint at all and what this would mean.
Suppose that R: |CAT| --> CAT is right adjoint to the quotient q. Be abuse, we will identify the objects of CAT and |CAT|. This means that for all categories X and Y we have:
$CAT(X, R(Y)) = |CAT|(X, Y) = |CAT|(X \times J, Y) = CAT( X \times J, R(Y))$
So $R(Y)$ must be cartesian local with respect to the map J --> pt. In particular we see, taking X=pt, that R(Y) has no non=identity automorphisms, i.e. R(Y) is rigid.
Now let's look at an example. Consider the group Z viewed as a category with one object. Then the endomorphisms of Z in |CAT| is the monoid Z (under multiplication). Now if R is fully-faithful then we need:
$ \mathbb{Z} = |CAT| (\mathbb{Z}, \mathbb{Z}) = CAT (R(\mathbb{Z}), R(\mathbb{Z}))$
but this last is equal to the automorphism of some set R(Z), hence we have a contradiction. So there is no such fully-faithful right adjoint.
In order not to have to worry about size issues, I'm going to answer the following question instead:
For a (small) cardinal number $\kappa$, is the category of small categories with $\kappa$-small 2-colimits 2-cocomplete?
If you take $\kappa$ to be inaccessible, then this will correspond to your question, under a particular choice of foundations. I presume moreover that you mean "2-colimits" in the weak "up-to-equivalence" sense which the nLab uses (which 2-category theorists traditionally call "bicolimits").
The fact that the 2-category Cat of small categories is 2-cocomplete, in this sense, has been well-known to category theorists for decades. It is obvious that Cat is cocomplete as a 1-category (since it is locally finitely presentable), and since it is closed symmetric (cartesian) monoidal, it follows by general enriched category theory that it is cocomplete as a category enriched over itself. In the nLab terminology, it has all strict 2-colimits. We then observe that strict pseudo 2-limits, which are 2-limits that represent cones commuting up to isomorphism but satisfy their universal property up to isomorphism (rather than equivalence), are particular strict 2-limits. Since any strict pseudo 2-limit is also a (weak) 2-limit, Cat is 2-cocomplete.
Now as Zoran pointed out in the comments, there is a 2-monad on Cat whose algebras are categories with $\kappa$-small colimits; let us call this 2-monad $T$. The strict $T$-morphisms are functors which preserve colimits on-the-nose, while the pseudo $T$-morphisms are those which are $\kappa$-cocontinuous in the usual sense (preserve colimits up to isomorphism). Therefore, the question is whether the 2-category $T$-Alg of $T$-algebras and pseudo $T$-morphisms is 2-cocomplete.
The answer is yes: it was proven by Blackwell, Kelly, and Power in the paper "Two-dimensional monad theory" that for any 2-monad with a rank (preserving $\alpha$-filtered colimits for some $\alpha$) on a strictly 2-cocomplete (strict) 2-category, the 2-category $T$-Alg is (weakly) 2-cocomplete. The 2-monad $T$ has a rank (namely, $\kappa$, more or less), so their theorem applies. I believe this all works just as well in the enriched setting.
Best Answer
I'm not entirely sure what you're looking for in an answer, but maybe I'll flesh out my comment.
It looks like what you're describing is equivalent to the homotopy category associated to the model structure on Cat where the weak equivalences are equivalences of categories. (I can say "the" because there is only one such, as pointed out in the comments. The cofibrations are functors injective on objects, and the fibrations are "isofibrations".)
I would say that in this context your category has been much studied. In particular, it is interesting to ask questions about homotopy limits and colimits in this category because many useful constructions arise in this way. (Homotopy (co)limits with this model structure are the same as "2-(co)limits" which is the name appearing in most of the literature, especially older literature.)
An example application of this language is the following theorem: The subcategory of presentable (resp. accessible) categories is closed under homotopy limits.
Using this one can prove that most of your favorite things are presentable (resp. accessible). For example, the category of modules over a monad arises via a homotopy limit construction, and this takes care of most things of interest.
Here's a neat application of this (which is the ordinary category version of a result that can be found, for example, in Lurie's HTT, 5.5.4.16.).
Say you want to localize a category $\mathcal{C}$ with respect to some collection of morphisms, $S$. Usually $S$ will not be given as a set, but if $\mathcal{C}$ is presentable you're usually okay if $S$ is generated by a set. Well, it turns out that if $F: \mathcal{C} \rightarrow \mathcal{D}$ is a colimit preserving functor between presentable categories, and $S$ is a (strongly saturated) collection of morphisms in $\mathcal{D}$ that is generated by a set, then $f^{-1}S$ is a (strongly saturated) collection of morphisms generated by a set. The argument goes by way of showing that the subcategory of the category of morphisms generated by $f^{-1}S$ is presentable, using a homotopy pullback square.
Adapting this to the model category or $\infty$-category setting, one sees immediately that localizing with respect to homology theories is totally okay and follows formally from this type of argument. (Basically, after fiddling around with cells to prove the category of spectra is presentable, you don't have to fiddle any more to get localizations. This is in contrast to the usual argument found in Bousfield's paper. You've moved the cardinality bookkeeping into a general argument about homotopy limits of presentable categories.)
Anyway, apologies for the very idiosyncratic application of this language; these things have been on my mind recently. I'm sure there are much more elementary reasons why one would care about using the model category structure on Cat.