Is there a more direct and elementary proof than Micheal ones, which uses Serre-Swan?

Thanks very much! After reading your solution I managed it to find a reference for your lemma. See Greub, Halperin, Vanstone: "Connections Curvature and Cohomology Volume I", Ch. II. 2.26 Prop. XVI.

Do you have some references for discussions of non-integrable physically relevant classical mechanical systems, where symplectic reduction brings some new insights?

Do you think it would be bring new insights to have quantization methods which work generally well for complicated geometries so that you are able to quantize the reduced space directly in physical relevant cases?

@Mariano Suárez-Alvarez, The point why I put the points 1) - 3) together was simply that for all three objects I am interested in a "list" or an overview of what is known about computed examples. The motivation for this question was that all three cohomologies occur in existence and classification results of what I am studying at the moment (star products, invariant star products, quantum momentum maps...). By the way, if $\mathfrak{G}$ is the Lie-Algebra of a compact Lie-group $G$ the Lie-algebra-cohomology is the same as the Lie-group-cohomology.

Are there special names in the literature for the situations in 1) and 2) I mentioned ? I have found something about 2), if the lie-Algebras arise as Lie-Algebras of some connected Lie-Groups, $G$ resp. $G_1$ ($G_1$ normal Lie-subgroup in $G$), having a vector space complement $\mathfrak{g}_2$ with $[\mathfrak{g}_1,\mathfrak{g}_2] = 0$ means that $\mathfrak{g}_2$ ist $G_1$-invariant. And such a homogenous space $G/G_1$ is called reductive, I have found that in O'Neill, Semi-Riemannian-Geometry and Baums "Eichfeldtheorie" book. But I havn't found any more comprehensive references.

Thanks! Do you have some references for what you added?