Is the Fundamental Group of space with contractible universal cover torsion-free?

This is false in general; in fact every group $G$ is the fundamental group of a unique (up to weak homotopy) space with contractible universal cover, namely the classifying space $BG$, or equivalently the Eilenberg-MacLane space $K(G, 1)$. In particular, the Klein bottle is such a classifying space, as is the torus.

What's true is the following.

Theorem: If $BG$ is a finite-dimensional CW complex, then $G$ is torsion-free.

Proof. With the above hypotheses, the universal cover of $BG$ is also a finite-dimensional CW complex on which $G$ acts freely, hence for any subgroup $H$ of $G$, the quotient $BH$ of the universal cover by the induced action of $H$ is also a finite-dimensional CW complex, and in particular has finite cohomological dimension. But finite cyclic groups have infinite cohomological dimension, by the standard computation of their group cohomology. $\Box$

No, the infinite sphere is the universal cover of $BZ/2$ the classifying space of $Z/2$.