dg.differential geometry - The invariant 3-form on a compact Lie group

Let $G$ be a compact Lie group. We have the well-known Maurer-Cartan left-invariant and right-invariant $1$-forms $theta$ and $bartheta$ in $Omega^1(G, mathfrak{g})$, probably discussed in every Lie theory lectures.

However the canonical bi-invariant closed $3$-form $chi = frac{1}{12} (theta, [theta, theta]) = frac{1}{12} (bartheta, [bartheta, bartheta])$ in $Omega^3(G)$ may be a little bit less-known. And when I heard of it I had some questions in mind...

1) Are there canonical invariant $5$-forms and higher invariant forms on a $G$?

2) How are they related to Lie algebra cohomology and equivariant cohomology?

3) The construction looks a little bit like $tr(A wedge dA)$, if we regard $theta$ as a connection $A$ on the frame bundle of $G$, and use the Maurer-Cartan equation $dtheta = -frac{1}{2}[theta, theta]$.

Now there is another famous $3$-form: the Chern-Simons form $tr(A wedge dA + frac{2}{3} A wedge A wedge A)$, and I wonder if these two are somehow related. Does the $tr(A wedge A wedge A)$ part vanish here, due to Jacobi identity?

(and, note there are Chern-Simons 5-forms etc.)

Thank you very much.