Two problems on the distribution of Carmichael's lambda function

Let $\lambda \left(n\right)$ denote the exponent of the multiplicative group modulo n. We show that when q is odd, each coprime residue class modulo q is hit equally often by $\lambda \left(n\right)$ as n varies. Under the stronger assumption that $gcd(q,6)=1$, we prove that equidistribution persists throughout a Siegel–Walfisz-type range of uniformity. By similar methods we show that $\lambda \left(n\right)$ obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order of a mod n, for any fixed integer $a\notin \{0,\pm 1\}$.