I recently got to know about the existence of the so-called Godbillon–Vey invariant, and I am interested in its relationship with foliation theory in 3-manifolds. I briefly recall here the definition:

Suppose that $\mathcal{F}$ is a codimension-$1$ foliation of a $3$-manifold $M$, and suppose that $\mathcal{F}$ is defined as the kernel of a $1$-form $\alpha$. We have that $d\alpha\wedge\alpha=0$ and this implies that there exists a $1$-form $\theta$ such that $d\alpha=\theta\wedge\alpha$ and now one can define the $3$-form $\theta\wedge d\theta$. This form is closed and its cohomology class does not depend on the choice of $\theta$, and it is called the Godbillon–Vey invariant of $\mathcal{F}$.

I got the idea that this invariant should tell us something about the transverse topology of the foliation, and I have some questions about it.

Does the vanishing of the Godbillon–Vey invariant imply something about the leaf space of $\mathcal{F}$? More precisely, can we say whether this space is Hausdorff or not?

Is there some known technique to calculate this invariant?

Is there some generalization of this invariant for foliation that are only $C^{\infty,0}$? That is to say, the leaves are smooth but the tangent plane bundle to the foliation is only continuous.

Any reference is appreciated.