2.Every maximal Cohen-Macaulay module is torsion-free. Here a proof attempt for $R$ a local CM ring with $K=R/m$, where $m$ is the maximal ideal of $R$.
Lemma: A noetherian module M is torsion-free if and only if $Hom_R(K,M)=0$.
(proof: The maps in $Hom_R(R/m,M)$ correspond to left multiplications $L_z$ with $z \in Hom_R(R,M)=M$ such that $z m=0$. )
Now assume $M$ is MCM, then $depth(M) \geq 1$ and thus $Hom_R(K,M)=0$, thus $M$ is torsionfree.
(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)