As pointed out by Moishe Kohan in the comments below, the following doesn't answer the question as asked, because my group $\Gamma$ is not contained in $SO(3,1)$.  Anyway, here is an easy description of a $\Gamma$ that contains orientation-reversing elements that has all the other properties that are asked for.  

By the intermediate value theorem, there is a regular dodecahedron in hyperbolic 3-space whose dihedral angles are all $90^\circ$.  (The angles for a Euclidean regular dodecahedron are greater than $90^\circ$ and the angles for an ideal regular dodecahedron are $60^\circ$.)  The group generated by the reflections in the sides of this dodecahedron is a discrete group $\Gamma$ with the property that $\mathbb{H}/\Gamma$ is contractible, since it's just a copy of the dodecahedron.