lists of computed cohomologies? Is there any comprehensive list of examples for computed
1) de-Rham cohomology-groups
2) Lie-algebra-cohomology groups $H^i(\mathfrak{g},\mathbb{R})$
3) equivariant de-Rham cohomology groups ?
Especially I am interested in a comprehensive list of examples (or classes of examples) of
a) manifolds with vanishing first and second de-Rham-cohomology
b) lie-algebras with vanishing first and second lie-algebra-cohomology
 A: You might be interested in the Equivariant cohomology wiki, which has a table of which cohomology theories have been computed for which symmetric spaces.
A: Consolidating the previous comments, one can start by understanding the topology of a compact simple group. Here is an example: the Lie group Sp(3) of dimension 21. It has rank 3 and the same Betti numbers as the product S^3 x S^7 x S^11 of spheres of dimension 2i+1; the numbers i (here 1,3,5) are the exponents of Sp(3). This theory dates back to Hopf, and was nicely explained by Bott using Morse theory. The (trivial) Lie algebra cohomology ring is generated by the associated forms in dimension 3,7,11. All this theory can be extended to compact irreducible symmetric spaces G/H, which (if not Hermitian) will also have b_1=b_2=0.
A: Chevalley and Borel calculated the cohomology of many Lie groups and Landweber collected those results here in just two pages: it describes the cohomology of $U(n)$, $SU(n)$, $SO(n)$, $G_2$, $F_4$, $E_6$, $E_7$ and $E_8$. 
You might also enjoy Neil Strickland's bestiary of topological spaces and spectra, which lists the (co)homology of many topological spaces.
A: You have to be a little bit more specific in your question (b), since Lie algebra cohomology groups are defined with respect to a module.
Indeed, there is a cohomological criterion for semisimplicity of (real) Lie algebras, which says that a (real) Lie algebra $\mathfrak{g}$ is semisimple if and only if for any finite-dimensional $\mathfrak{g}$-module $\mathfrak{M}$, $H^1(\mathfrak{g},\mathfrak{M})=0$.
This would seem to answer your question (b) if you mean vanishing in this strong sense.  If, on the other hand, you mean vanishing of $H^i(\mathfrak{g},\mathbb{R})$ for $i=1,2$, with $\mathbb{R}$ the trivial one-dimensional module, then I am not aware of any very general results.  The vanishing of $H^1(\mathfrak{g},\mathbb{R})$ says that $[\mathfrak{g},\mathfrak{g}] = \mathfrak{g}$ and such Lie algebras are called perfect.  Such algebras cannot be solvable, but there are non-semisimple examples.
