Question. Is it true that each infinite hyperbolic group has a torsion-free subgroup of finite index?

Are there counterexamples, or positive results for some large subclasses of hyperbolic groups? For example, is the answer positive for orbifold fundamental groups of negatively curved orbifolds? More precisely, I am interested the most in the case of orbifolds with locally $CAT(0)$ metric. I guess it will be hard to construct a counterexample in this category.

Related question. Is it known that every nontrivial hyperbolic group has a proper subgroup of finite index?

Just to recall, a definition of hyperbolic group is here https://en.wikipedia.org/wiki/Hyperbolic_group .

Added. Note, that every hyperbolic group is finitely presented (thanks to Sam Nead).