**Question 1:** The set is dense.  

Suppose that we are given a fixed $x\in\mathbb{R}$.  Then let $p$ be a large prime.  If $p$ is sufficiently large, then there will be a prime $$q\in\left[px,\ px+\left(px\right)^{0.525}\right]$$ by [the work][1] of Baker, Harman and Pintz on [prime gaps][2].  This implies that $$\left|x-\frac{q}{p}\right|\ll_x p^{-0.475},$$ which becomes arbitrarily small as we take $p\rightarrow\infty $.  This proves that for any $\epsilon>0$, there exists $p,q$ such that $\left|x-\frac{q}{p}\right|\leq \epsilon.$


**Question 2:**  We can find infinitely many solutions to $$1\leq qp-rs\leq a$$ for primes $p,q,r,s$ and all $a\geq 26$.  Under the [Elliott-Halberstam Conjecture][3], we can take $a\geq 6$.


This is a corollary of [the work][4] of Goldston, Graham, Pintz and Yıldırım on the gaps between almost primes.  They prove that if  $q_n$ is the $n^{th}$ almost prime, then $$\liminf_{n\rightarrow \infty} q_{n+1}-q_n \leq 26,$$ and that the upper bound may be reduced to $6$ under the Elliott-Halberstam Conjecture.  Since $q_n=pq$ and $q_{n+1}=rs$ where $p,q,r,s$ are primes, this yields the above claim.


  [1]: http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf
  [2]: http://en.wikipedia.org/wiki/Prime_gap
  [3]: http://en.wikipedia.org/wiki/Elliott%25E2%2580%2593Halberstam_conjecture
  [4]: http://arxiv.org/abs/math/0506067