2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)