  Consider the random Cayley graph of a finite group $G$ with respect to $k$
generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The
results of this article supplement those in the three main papers on random
Cayley graphs. The majority of the results are inspired by a `universality'
conjecture of Aldous and Diaconis (1985).
  To start, we study the limit profile of cutoff for the simple random walk on
this random graph, as well as a detailed investigation into mixing properties
when $G = \mathbb Z_p^d$ with $p$ prime. We then exposit a proof of Diaconis
and Saloff-Coste (1994) establishing lack of cutoff when $k \asymp 1$. We move
onto discussing material from our companion paper on matrix groups. We then
study distance of a typical element of $G$ from the identity in an $L_q$-type
graph distance in the Abelian set-up. Finally, we give necessary and sufficient
conditions for $k$ independent uniform elements of $G$ to generate $G$, ie for
the random Cayley graph to be connected, based on work of Pomerance (2001).
  The aforementioned results all hold with high probability over the random
Cayley graph.
