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.