It's harder to extend etale cohomology, since you might not have enough etale maps from schemes (unless you're Deligne-Mumford). Nevertheless, one can make sense of a derived category of etale sheaves together with the various usual functorialities (including the Grothendieck-Lefschetz trace formula!). A much harder question is to extend notions related to algebraic cycles and intersection theory. This has been intensely studied via things like Gromov-Witten theory. (I've never seriously thought about differentiable stacks; in principle mostly everything should carry over)

Have you looked at the Stacks project? The chapters on algebraic stacks collect this. It's a long list. All the basic notions go over: separated, finite type, proper, smooth, dimension, normal, flat... You can extend quasicoherent sheaf cohomology by descent.

"Nice" stacks over an algebraically closed field $k$ are locally of the form $[X/G]$ for an affine scheme $X$ and a linearly reductive $k$-group scheme $G$: see arxiv.org/abs/1504.06467. You can picture this by drawing the $G$-orbits on $X$ and decorating points with the stabilizers.

Not relevant to your actual question, but why do you say that we needed things like the pro-etale topology before this perspective on etale cohomology was possible? Why are the traditional foundations for lisse sheaves fine for constructing a derived category with good properties, but not for the lift to a DG category? (Also, the Artin-Mazur etale homotopy type should certainly give a legitimate DG algebra structure on etale cohomology).

YCor's argument works in greater generality, but in your case you can get an explicit bound. If $A = k[x_1,\ldots,x_n]/I$ is finitely generated over a field, the Zariski tangent space at any point of $\mathrm{Spec}(A)$ has dimension at most $n$. This is because the tangent space of a point of $\Spec(A)$ is canonically a subspace of the tangent space of the corresponding point in $\mathbf{A}^n$.

This is crucial when studying the geometry of the moduli stack $\mathrm{Bun}_G(X)$ of $G$-bundles on a curve $X$.

Yes, I think both of those adjustments are right. Thanks for the reference! I'm surprised by the simplicity of the argument - it looks like Katz essentially just writes down a solution directly as a "matrix exponential". The condition of being $(t_1, \ldots, t_n)$-adically continuous should say exactly that the integrable connection is specified by differential equations $\frac{\partial}{\partial t_k} s = \sum_j M^j_{ik} s$, right? (i.e. some random integrable connection on the power series ring might not land in the span of the $dt_i$'s.)