Not a complete answer, but a bit too long for a comment: Conjecture 1 is very likely to be very difficult if true. The corresponding conjecture for general primes is open. Let $p_n$ be the $n$th prime and $g_n$ be the $n$th prime gap. If one has $\sqrt{p_{n+1}}-\sqrt{p_n} <1 $ for sufficiently large primes then one would have that $g_n = O(p^{1/2})$. The  best current result (as far as I'm aware) of that form we can actually prove is that $g_n = O(p^{\frac{3}{4} + \epsilon})$ for any $\epsilon>0$ (due to Nikolai Chudakov).  Your conjecture would imply that $g_n =  O(p^{1/2})$ and is seems to be substantially stronger. Since we have that $R_n$ is in general very close to $p_{2n}$ your conjecture isn't implausible, but likely to  be stronger than what we can currently prove.