I'm having a hard time trying to provide an example of the following two problems:
An example of a ring $R$ and an ideal $I$ of $R$ such that neither $R$ nor $I$ has zero divisors but the quotient group $R/I$ does.
A ring $R$, an ideal $I$ of $R$, and an element $r$ in $R - I$ such that $a$ is not idempotent in $R$ but $a + I$ is idempotent in the quotient group $R/I$.
For 1. take $R$ to be $\mathbb Z$ the ring of integers and $I$ to be $6\mathbb Z$. Then $[2],[3]$ are zero divisors in $\mathbb {Z/6Z}$
For 2. take the same $R$ and $I$ then $[4]^2=[4]$ but $4$ is not idempotent in $\mathbb Z$ (same with $3$ and $[3]$)