Let $F$ be a c.d.f. over reals and $\{R_i\}$ be the set of all real random variables with c.d.f. $F$. Do all $R_i$ have the same expected value? i.e. is the following statement true?
\begin{equation} \mathbb{E}[R_i] = \mathbb{E}[R_j] \quad \forall i, j \end{equation}
If yes, why? If not, could you show me a counter-example?
Yes, and you do not need the distribution to be continuous or have a density. You can say
$$\mathbb E[R_i] = \int\limits_0^{\infty}(1- F(x))\, dx-\int\limits_{-\infty}^0 F(x)\, dx$$ if the two integrals are finite; if both are infinite then the expectation is not defined.
So the same cumulative distribution function gives the same expectation.