Can the factor ring $I/J$ be expressed in terms of sums, quotients or submodules of rings of the form $R/K$, where $K$ is any left ideal of $R$?
The most sensible correction to this I can see is
Can the factor ring $J/I$ be expressed in terms of sums, quotients or submodules of modules of the form $R/K$, where $K$ is a left ideal of $R$?
Notice that I had to switch "rings" to "modules" because you required $K$ to be a left ideal, but $R/K$ won't be a ring unless it is a two-sided ideal.
In this interpretation, $J/I$ is a submodule of $R/I$.
It's unclear what definition of torsion you wanted, but you haven't clarified for over a year, so I'll give it a shot.
Certainly $J/I$ does not have to be a singular module, and it certainly might have a nonzero element with annihilator zero, so it would not be torsion in either of these senses.