Categorical mapping class group action

[This is an elaboration of parts of Mark Penney's answer]

A natural source of categorical actions of the mapping class group is the category assigned by any 4d TFT to a surface. Such categories are often written as Fukaya categories - eg Donaldson theory and Seiberg-Witten theory will attach Fukaya categories of moduli of G-bundles or symmetric products of the curve - but there are other examples, coming e.g. from factorization homology constructions as Mark wrote.

Here's what seems like the most natural example: the MCG acts on moduli spaces of G local systems ("character varieties" - or really character derived stacks) for G a complex reductive group say. So it acts on any category of sheaves on these moduli spaces. One natural answer is to look at quasicoherent sheaves - i.e. to any $\Sigma$ we look at $QC(Loc_G\Sigma)$. This is neither smooth nor proper as dg category, but it's faithful and interesting.. It's also the output of factorization homology, when the input is the braided (in fact symmetric) monoidal category $Rep(G)$ of representations of $G$. A more interesting answer coming from [Betti version of] geometric Langlands is to replace $QC$ by $IndCoh$ (possibly with fixed singular support). This is supposed to indeed be the value of an interesting 4d TFT on $\Sigma$ - namely the Kapustin-Witten B-twist of $\mathcal N=4$ super Yang-Mills. [An even more interesting (but conjectural) answer is the A-side of geometric Langlands.]

The character varieties have a symplectic form preserved by $MCG(\Sigma)$, so one can try to quantize them and get more interesting categorical actions. That's what is done e.g. in my work with Brochier and Jordan that Mark referenced, using factorization homology.

As a (partial) answer to question 2, factorization homology provides examples of objects carrying actions of the mapping class groups. I will follow the notation of the paper Factorization homology for topological manifolds by Ayala-Francis avaiable at https://arxiv.org/abs/1206.5522, so see there for more details.

We start with a symmetric monoidal $\infty$-category $\mathcal{C}$ such that the underlying category is presentable and the product distributes over colimits. Then the short story is that factorization homology takes as input a $\mathrm{Disk}_2^\mathrm{or}$-algebra $A$ in $\mathcal{C}$ and produces a symmetric monoidal functor $\int_A: \mathrm{Mfld}_2^\mathrm{or} \to \mathcal{C}$. In particular, for each surface $\Sigma$ the object $\int_A \Sigma \in \mathcal{C}$ has an action of $\mathrm{Aut}(\Sigma) = \mathrm{MCG}(\Sigma)$.

A $\mathrm{Disk}_2^\mathrm{or}$-algebra is defined in the following way: Let $\mathrm{Mfld}_2^\mathrm{or}$ be the symmetric monoidal $\infty$-category having as objects oriented $2$-manifolds (admitting finite good covers) and morphism spaces orientation preserving embeddings. The symmetric monoidal structure is given by disjoint union. Define $\mathrm{Disk}_2^\mathrm{or}$ to be the full subcategory of $\mathrm{Mfld}_2^\mathrm{or}$ on those objects whose underlying manifold is a finite disjoint union of $\mathbb{R}^2$. Then a $\mathrm{Disk}_2^\mathrm{or}$-algebra in $\mathcal{C}$ is simply a symmetric monoidal functor $A: \mathrm{Disk}_2^\mathrm{or} \to \mathcal{C}$. Note that this is equivalent to being an algebra over the framed little 2-disks operad of Salvatore-Wahl.

I can't say much about dg- or $A_\infty$-categories, so instead let's consider a simpler setting. Let $\mathrm{Rex}$ be the symmetric monoidal category of finitely cocomplete $\mathbb{C}$-linear categories and right exact functors between them with the Deligne-Kelly tensor product $\boxtimes$. Then the symmetric monoidal $\infty$-category $\mathcal{C}$ obtained by localising $\mathrm{Rex}$ at the categorical equivalences satisfies the properties we need. $\mathrm{Disk}_2^\mathrm{or}$-algebras in $\mathcal{C}$ correspond to what are known as balanced tensor categories.

Examples of balanced tensor categories include the categories of representations of the quantum groups $U_q\mathfrak{g}$, or more generally of ribbon Hopf algebras. Ben--Zvi-Brochier-Jordan have a project studying factorization homology of balanced tensor categories with particular focus on quantum group representations. I suggest you have a look there for interesting examples of linear categories carrying mapping class group actions.