I found the introduction of this article of A. Furman (mainly quoting results of Furstenberg) quite useful in summarizing the big picture:

http://homepages.math.uic.edu/~furman/preprints/fb.pdf

In particular, you can define the universal boundary (boundary = compact Hausdorff $G$-space such that the action is minimal and strongly proximal) for any locally compact group $G$, and $G$ is amenable if and only if the universal boundary is trivial.  It looks like this universal boundary is what is usually meant by the Poisson-Furstenberg boundary of a locally compact group.

As far as I know, if the group has exponential growth, that in itself doesn't tell you anything about the boundary.