$\DeclareMathOperator\Map{Map}\DeclareMathOperator\ad{ad}$This is an attempt to answer the third question: what else?
Let $X$ be a compact space and let $G$ be an affine algebraic group. One can contemplate the following (underived) higher stacks:
$BG$: the classifying stack of $G$-torsors.
$X_B$: the constant stack associated to $X$.
One can consider the higher underived mapping stack $\Map(X_B,BG)$, which is nothing but the Deligne–Mumford stack of $G$-local systems on $X$. Its tangent complex has amplitude $[-1,0]$:
in degree $-1$, at a $k$-point $P$ ($P$ is a $G$-local system), its cohomology is $H^0(X,\ad(P))$, where $\ad(P)$ is the linear local system associated with $P$ and the adjoint $G$-representation $\mathfrak{g}$: $\ad(P)=P\times_G\mathfrak{g}$.
in degree $0$, at a $k$-point $P$, its cohomology is $H^1(X,\ad(P))$.
The infinitesimal theory $\Map(X_B,BG)$ doesn't capture anything about higher cohomology groups $H^{*\geq 2}(X,\ad(P))$.
If you're looking at the derived mapping stack $\mathbb{R}{\Map}(X_B,BG)$ instead, then its tangent complex at a $k$-point $P$ is the full de Rham cohomology $H^{*+1}(X,\ad(P))$.
Why is this so? The point is that the underived stack $\Map(X_B,BG)$ doesn't see anything else than the fundamental groupoid of $X$: this is because $BG$ is a $1$-truncated Artin stack. But if we allow families of $G$-local systems parametrized by geometric objects intrinsically carrying homotopical information (affine derived schemes), then we get back the missing information. For instance, it's a good exercise to check that if $Y$ is the derived self-intersection of $0$ in the affine line that was mentionned in previous answers (i.e. $Y=\operatorname{Spec}(k[\tau])$, $\operatorname{deg}(\tau)=-1$), then a $Y$-point in $\mathbb{R}{\Map}(X_B,BG)$ is the datum of a $k$-point $P$ and a class in $H^2(X,\ad(P))$.
