Thank you for the reference!

What I did not see, and don't know about at all, are ways to guarantee that H^3 vanishes. I don't know if, by analogy, this could some times be implied by some assumptions on symmetric spaces, be it compactness or some sort of connectedness (perhaps 3-connectedness in this case?).

@Amr That's an interesting question (to me at least)! I'm not very familiar with the setting but I looked a bit into what I could find in the literature. There is a cohomology (Yamaguti cohomology) for Lie triple systems (Lts) that when using the Lts itself as the module (analogous to the adjoint for Lie algebras) controls the deformations of the Lts. Kubo, Taniguchi, A controlling cohomology of the deformation theory of Lie triple systems If $H^3$ vanishes, then the Lts is rigid in some sense, see theorem 2 in that paper.

@MikhailBorovoi let me give a Mathoverflow reference for this, because it has both references for the result, some proofs right there, and a lot of interesting discussion (in my view): mathoverflow.net/q/8957/104042 . I think I was wrong about attributing it to Hopf, what I had in mind his result only implies finite $\pi_2$, as explained in this answer: mathoverflow.net/a/8996/104042 . I don't know who first proved it. Some of the references there are "Representations of compact Lie groups" by Bröcker and tom Dieck and Milnor's book on Morse theory.

A Lie groupoid determines the singular foliation on $M$ and the classifying maps $f_i$, but this data does not determine the Lie groupoid. For example, a vector bundle can be seen as a Lie groupoid, (a bundle of abelian groups). The leaves $F_i$ are the points of $M$, and thus contractible. So the homotopy classes $f_i$ are the same for different vector bundles, i.e., for different groupoids. That data determines the foliation and s-fibres, but ignores how the s-fibres fit together geometrically. It is not a problem in the transitive case

Spaces modelled in arbitrary subsets of Euclidean spaces, where one tries to still do some meaningful differential geometry (differentiable functions, vector fields, flows, etc.), bring to mind subcartesian spaces, a particular kind of Sikorski spaces (also called differential spaces). If it is fine to restrict to spaces locally modelled on closed subsets of euclidean spaces (as objects made of wires, plates, balls likely are) then you could look at differentiable spaces (something like a particularly nice differential geometric scheme), which may be easier to manage.

@Z.M So maybe some careful restriction of the equivalence from Koszul duality is what applies here. Thanks for the reference! This paper may also be useful along those lines: Joost Nuiten, Koszul duality for Lie algebroids. But even in differential geometry people do consider more general, more algebraic versions of these ruths (and of Lie algebroids), for example: Luca Vitagliano, Representations of Homotopy Lie-Rinehart Algebras

@Z.M Indeed, a representation up to homotopy (ruth) of a Lie algebroid $A$ is a DG-module of the de Rham DGA of $A$, $\Omega^\bullet(A)$. To be precise (hope I don't sound nitpicky, not my intention), in the the differential geometric setting we ask it to be of a special kind: the underlying graded module is $\Omega^\bullet(A;E^\bullet)$ ($A$-forms with values on sections of the chain complex of vector bundles $E^\bullet$). Similar to how vector bundles over $M$ correspond to finitely-generated projective $C^\infty(M)$-modules.

A recent one for groupoids: Fernando Studzinski, On the cohomology of representations up to homotopy of Lie groupoids

As for PhD thesis with details, there are a few, I'll mention 1 early, 2 recent ones: the first one (groupoids, algebroids) would be the one of Camilo Arias Abad: Representations up to homotopy and cohomology of classifying spaces A recent one for algebroids, higher versions. Looking at it, the definition you wanted is on the bottom of page 17: Theocharis Papantonis, $\mathbb{Z}$-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures

Alternatively, the viewpoint usually taken for ruths: an $A$-connection on $E$ is the same as a degree 1 graded derivation $d$ on the $\Omega(A)$-module $\Omega^\bullet(A;E)$. It is flat (i.e. a representation of $A$) if and only if $d^2=0$. This also makes sense for a complex $E^\bullet$, it is just that the grading on $\Omega^\bullet(A;E^\bullet)$ now takes into account the grading on $E^\bullet$. And then $d^2=0$ if it is a ruth. That's the definition of A-superconnection in the paper of Gracia-Saz and Mehta, it is just that they are dealing with a 2-term complex.

I don't remember if I have seen explicitly the definition of $A$-connection on a chain complex singled out. Usually for A-connections on a complex, people are in the context of representations up to homotopy (ruths), and then simply present the connections on each bundle, the relations they satisfy are included in the conditions for being a ruth. You can find a good reference here (in particular points 1,2,8 of section 2 for what you want in the 2-term case): Frejlich, Pedro, Morita invariance of intrinsic characteristic classes of Lie algebroids link