**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][1], we can take $a\geq 6$.


This is a corollary of the [work][2] of Daniel A. Goldston, Sidney W. Graham, Janos Pintz and Cem Y. Yıldırım on the gaps between semi 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.


  [1]: http://en.wikipedia.org/wiki/Elliott%25E2%2580%2593Halberstam_conjecture
  [2]: http://arxiv.org/abs/math/0506067