With regards to Q1 (and part of Q4), the numbers of the form $R(G)$ are dense in $\mathbb{R}_{\ge 0}$ even when $G$ is restricted to be abelian.

Some first results. $R(G \times H) = R(G) R(H)$ if $\gcd(|G|, |H|) = 1$. We also have $R(C_p) = \frac{2}{p}$ and $R(C_p^2) = 1 + \frac{3}{p}$. 

**Lemma:** Let $a_1, a_2, ... $ be a sequence of positive reals such that $\lim_{n \to \infty} a_n = 0$ but such that $\sum a_n$ diverges. Then the set of sums of finite subsequences of the $a_i$ is dense in $\mathbb{R}_{\ge 0}$.

_Proof._ Let $r \in \mathbb{R}_{\ge 0}$ and fix $\epsilon > 0$. Choose $N$ such that $a_n < \text{min}(r, \epsilon)$ for all $n \ge N$. Then the partial sums starting from $a_N$ diverge but begin less than $\text{min}(r, \epsilon)$ and increase by at most $\epsilon$ at each step, so the conclusion follows. $\Box$

Applying the lemma to the sequence $a_n = \log \left( 1 + \frac{3}{p_n} \right)$ where $p_n$ is the $n^{th}$ prime, we conclude that the numbers of the form $R(G)$ where $G$ is a product of groups of the form $C_p^2$ for distinct primes $p$ are dense in $\mathbb{R}_{\ge 1}$, and if we allow in addition the groups of the form $C_p$, then the conclusion follows.

Q2 appears to be potentially very difficult and I have not thought about Q3.