A model category of abelian categories? Let $\mathcal{M}$ be the following category:


*

*The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels.

*The morphisms are functors that preserve the structure strictly.

*Composition and identities are the obvious ones.


If I'm not mistaken, $\mathcal{M}$ is a locally finitely presentable category. At any rate, the forgetful functor $U : \mathcal{M} \to \mathbf{Cat}$ preserves all limits and filtered colimits, so under the assumption that $\mathcal{M}$ is l.f.p., we have a left adjoint $F : \mathbf{Cat} \to \mathcal{M}$.
Question 1. Can the standard model structure on $\mathbf{Cat}$ be transferred to $\mathcal{M}$, i.e. does there exist a model structure on $\mathcal{M}$ where the weak equivalences and fibrations are created by $U : \mathcal{M} \to \mathbf{Cat}$?
Question 2. Assuming the desired model structure on $\mathcal{M}$ exists, is it compatible with the obvious $\mathbf{Cat}$-enrichment, i.e. do we have a model 2-category?
Question 3. Let $\mathfrak{M}$ be the 2-category whose objects and morphisms are as in $\mathcal{M}$ and whose 2-cells are natural transformations; and let $\mathfrak{A}$ be the 2-category whose objects are small abelian categories, whose morphisms are exact functors, and whose 2-cells are natural transformations. There is an evident 2-functor $\mathfrak{M} \to \mathfrak{A}$ which is surjective on objects, (locally) injective on morphisms, and (locally) bijective on 2-cells. Does this exhibit $\mathfrak{A}$ as the higher-categorical localisation of $\mathfrak{M}$ with respect to weak equivalences?
 A: It seems to me that you can answer this question without 2-monads.  Perhaps 2-monads do apply in this example but they might not in similar ones so here is an outline not using them.
(1)Yes.  More generally consider an adjunction $F\dashv U:M \leftrightarrows Cat$ with M locally presentable as a 2-category (just amounts to its underlying category being locally presentable and it has cotensors with the walking arrow 2 - equally it is complete as a 2-category) and U an accessible right 2-adjoint (just means its underlying functor is accessible right adjoint and preserves cotensors with 2, equally all limits).
Then you get a projectively lifted model structure in which the weak equivalences and fibrations are those whose image under U is one.
Now M, being complete, has cotensors with the free isomorphism $I$.  The objects $X^{I}$ provide functorial path objects for $M$: indeed you get a factorisation of $X \to X \times X$ into an internal equivalence followed by a (discrete) isofibration in any 2-category.  So according to Proposition 1 on
http://ncatlab.org/nlab/show/transferred+model+structure
you get a transferred model structure.
(2) I haven't checked but I would strongly suspect so.  Certainly the relevant condition is true for generating cofibrations and trivial cofibrations (by adjointness) and presumably you can extend from there by general nonsense?
(3) This can be seen in a couple of ways.  As David Roberts says, you can use Pronk's work on bicategorical localization, and that might be the most direct route.  The key point here is that given any weak map $f:A \rightsquigarrow B$ you can cover it by a span of strict maps $(p,q):A \leftarrow P_{f} \rightarrow B$ where $p$ is a trivial fibration and $q$ an isofibration such that $fp \cong q$.  Here $P_{f}$ is the pseudolimit of the arrow $f$ - in the $F$-categorical sense of Lack and Shulman.  It is just the full subcategory of the comma category $B/f$ consisting of the invertible arrows, with the evident structure lifted to $M$.
In particular this shows that the inclusion $M \to \mathbf M$ is sufficiently surjective on 1-cells that Pronk's Proposition 24 may be applied.  Though you would also need to verify the calculus of fractions stuff therein to apply that too.
Beyond this I would point out that a 2-functor (or pseudofunctor) $F:M \to C$ sends weak equivalences to equivalences iff it sends trivial fibrations to equivalences (use the above covering $(p,q)$ of a $U$-equivalence $f$ by a span of trivial fibrations, just like in Ken Brown's lemma). So $\mathbf M$ is equally the 2-categorical localisation at the trivial fibrations.
(3*)
Although this property characterises $\mathbf M$ up to biequivalence, there is in fact a stronger property characterising it up to isomorphism.  Namely, it is the (2)-category of weak maps for the algebraic weak factorisation system generated by the lifted generating cofibrations.  You can see this using an argument identical similar to the proof of Theorem 16 of http://arxiv.org/abs/1412.6560
From that you get a Kleisli 2-adjunction $Q \dashv j:M \to \mathbf M$ whose counit $QA \to A$ is a trivial fibration.  (Actually the awfs stuff only gives you the 1-adjunction, but since j preserves cotensors this extends to a unique 2-adjunction).
And you could now use an argument identical to that of Theorem 4.15 of the paper of Steve Lack mentioned by Alexander Campbell to verify the 2-categorical localisation property of Q3.
A: Expanding on my comment, there ought to be a finitary 2-monad $T$ on Cat with $\mathfrak{M} = T\text{-Alg}_s$ and $\mathfrak{A} = T\text{-Alg}$. If this is so, then all of your questions are answered in the affirmative by Steve Lack's paper Homotopy-theoretic aspects of 2-monads (arXiv link): Q1 and Q2 by Theorem 4.5, and Q3 by Theorem 4.15.
