Let $G$ be a permutation group of degree n, and let $r>1$ be the minimal block size of $G$. Also, let $s:=n/r$, so that G may be viewed as a subgroup in the wreath product $R\wr S$, where $R\le Sym(r)$ is primitive, and $S=\pi(G)\le Sym(s)$ is transitive ($\pi$ here denotes projection $G\rightarrow Sym(s)$). Then
$$d(G)\le \frac{a(R)bs}{\sqrt{\log_{2}{s}}}+d(S) \text{ ($\ast\ast$})$$
where b is an absolute constant, and $a(R)$ denotes the composition length of R. By a result of Pyber and Guralnick, $a(R) \le C\log_{2}(r)$ for an absolute constant $C$. Also, as you mentioned, $d(S)\le cs/\sqrt{\log_{2}{s}}$, for an absolute constant c. (For information on the constants $b,c$, and $C$, and on the bound at ($\ast\ast$), see http://arxiv.org/abs/1504.07506 .)

Now, we also have $\log{|G|}\le \log{|R\wr S|}\le rs\log{r}+s\log{s}$. This means that your first claim follows when $r>\sqrt{\log{s}}$, for example. Thus, it only remains to deal with the case $r\le \sqrt{\log{s}}$.

As you said, the key seems to be reducing the upper bounds when S is either $Alt(s)$ or $Sym(s)$. However, I think that a bound of the form $d(G)\le c'd(R)+2$ would be difficult to prove in this case, and is probably not true. For example, when $R=C_2$ and $K$ is the intersection of $G$ with the base group of the wreath product, then the number of generators for $K$ as an $S$-group can grow as $bs/\sqrt{\log{s}}$, where b is as above (see Section 3.1 of the paper I mentioned above).

**EDIT:** @Jan-Christoph Schlage-Puchta The first part of your question, that is, that for a transitive permutation group of degree $n$, $\frac{d(G)\log{|G|}}{n^{2}}$ tends to $0$ as $n$ tends to $\infty$, has now been proved in http://arxiv.org/abs/1601.02561 .