This set of questions pertains to the proof of Proposition 3.3.18(b) in Bruns and Herzog, Cohen-Macaulay Rings:
Question 1: It seems to me that under the hypothesis (a) of the theorem, the torsion-freeness of $\omega_R$ forces the underlying ring $R$ to be an integral domain. Is that correct or am i missing something? Here is my argument: We have that $\omega_R$ is torsion-free, so let $\xi \in \omega_R$ be a non-zero element. Now assume that $R$ is not an integral domain. Then there exist nonzero elements $a,b \in R$ such that $ab=0$. Since $b \neq 0$ and $\xi$ is not torsion, we must have that $b \xi \neq 0$. Since $0 \neq b \xi \in \omega_R$, $\omega_R$ is torsion-free and $a \neq 0$ we must also have that $a (b \xi) \neq 0$, which is a contradiction.
Question 2: I am puzzled by the inequality $\operatorname{depth} (R_p/\omega_R R_p) \ge \operatorname{depth}(R_p) -1$. How did the authors arrive to this? The way i understood this part of the proof is by noticing that $\omega_R R_p$ is a principal ideal generated by an $R_p$-regular element and so $R_p/\omega_R R_p$ is Cohen-Macaulay, which also shows that $R/\omega_R$ is CM.