Two problems on the distribution of Carmichael's lambda function

Let $\lambda(n)$ 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 $\lambda(n)$ 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 $\lambda(n)$ obeys Benford's leading digit law with respect to natural density. Moreover, if we assume GRH, then Benford's law holds for the order of $a$ mod $n$, for any fixed integer $a\notin \{0,\pm 1\}$.