Your limit should be a one instead of a zero. Let me also add that the result may in fact be over a given range of parameters of the in the definition of strongly regular graphs. The analogue for regular graphs would be: for each fixed $d$, almost all $d$-regular graphs contain a Hamilton cycle. In other words, when studying regular graphs, "almost all" statements often are proved after having fixed the parameter(s) rather than over all possible parameters.

What is the vertex ranking chromatic number? Could you add this to your question? Thanks!

That is a wonderful video!

Yes it is roughly unimodal. On a crude scale (p=c/n), it is logarithmic and increasing in c for c < 1, it jumps to order $n^{1/3}$ for $c=1$, and then it drops back to logarithmic and decreasing for $c>1$. If you parameterize $p$ more finely near $1/n$ you see more interesting behaviour emerge. Key papers by Luczak in the barely-below-(1/n) case, and by Riordan and Wormald (arxiv.org/abs/0808.4067) and Ding, Kim, Lubetzky and Peres (tinyurl.com/supdiam) in the barely-above-(1/n) case.

My answer to this question (mathoverflow.net/questions/38305/similarity-of-weighted-gra‌​phs/…) is also applicable to your question, in case that answer is useful to you.

It isn't? News to me.

I think this is a great question and have more than once wished for such a list in the past.