Models for computing cohomology of Lie groupoids

Question

Given a Lie groupoid $\mathcal{G}=[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$, let $\mathcal{G}_\bullet$ be the associated simplicial manifold.

Let $\Omega^\bullet(\mathcal{G}_\bullet)$ be the associated double complex. The $k$-th cohomology of this double complex is defined to be the $k$-th deRham cohomology of the Lie groupoid $[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$. This I am calling the simplicial model for the cohomology. Please correct me if there is already a name for this model.

Questions:

1. Where did the notion of deRham cohomology group(s) of a Lie groupoid appear for the first time?
2. To compute equivariant cohomology of a topological space/manifold $M$ with an action of a topological/Lie group, there are at least three models Weil model/Borel model/Cartan model. Are there other models that compute cohomology of Lie groupoids?
3. Are there other cohomology theories in the category of Lie groupoids?

Answer 1

This is 2 years late, but since nobody has answered I'll give it a go.

$1.$ I don't know the answer to this.

$3.$ Yes, you can take cohomolgy of Lie groupoids with respect to sheaves, just like you could for manifolds. You can take cohomology of $\mathcal{O}\,,\mathcal{O}^*\,,\mathbb{Z}$ for example. You can also take cohomology with respect to representations. Analogues of these cohomologies exist for Lie algebroids as well, and there is a map, called the van Est map, which relates them.

$2.$ The models you named can only be used to compute cohomology with respect to certain sheaves (like the constant sheaf $\mathbb{R}\,,$ or de Rham cohomology). In general the cohomology of Lie groupoids is defined by forming the nerve and taking its cohomology, as a simplicial space. This can be difficult to compute, but one way of doing it is by using van Est theorems.