Actually, I looked at Spitzer's book and the criterion you give is only stated for symmetric $\mu$. So again I'm not sure what the answer is.

Note: I added an additional hypothesis, namely that $\nu$ is finitely supported. Given this, it seems that your answer would imply that $\alpha \mu + (1-\alpha)\nu$ is also recurrent, given that $|n|^2\mu(n) = c+o(1)$. No?

But what if mu doesn't have a first moment? Is it obviously transient?

Thanks again Anthony. And yes - my choice of title is a little confusing.

Thanks! For amenable groups (at least) it seems like it should be possible to take the square root of i.i.d. actions. I wonder if more can be said.

Is it uniquely determined by the sizes of its intersections with each conjugacy class?

Sure! That would be perfect.

You mean k < n?

That would be great.