Edit: Erdős got three things wrong. First of all, it wasn't Faudree, Shelp, and Rousseau, it was Faudree, Shelp, and Burr. Second, it wasn't "recently", it was in the future (with respect to the quote you provide)! Third, they didn't prove that $(R(n+1,n)-R(n,n))/n \to \infty$, but only that $R(n+1,n)-R(n,n) \geq 2n-3$.
The relevant paper is On the difference between consecutive Ramsey Numbers, published in 1989. The proof is not long. On a (somewhat cursory) search I wasn't able to find any papers citing this one that address the same question, so it seems likely that this bound is still the best known.
For your first question: thanks to Miklos Simonovits and others, all of Erdős' papers are available from this site. I scanned through the papers by Erdős, Faudree, Rousseau and Schelp from up to 1982 but didn't see such a result. There are 12 papers with precisely these four coauthors, and another several that also have Burr as a coauthor, so it may take some time to find (especially if it isn't explicitly stated as a lemma but is embedded in a proof somewhere). But: if they published it, then you'll be able to find your result by scanning through the papers on that site.