Molecular mechanics, which is interested in simulation of big biomolecules and the like, ignore quantum mechanical effects, or use a hybrid of quantum/classical mechanics, because the structures of interest are large enough the QM effects can be approximated away.
Other subfields of computational chemistry use only QM, with some approximations to make the programs tractable.
I can't think of a particular example for an unknown reaction being predicted first, then verified, but it wouldn't surprise me. However, there are cases of computations being used to guide experiment (who have to have an idea of where to look for something).
Theory is generally used to explain, rather than predict. Ie, given a reaction, what is the most likely mechanism? Or helping explain the source of a particular property.
With increases in computing power, databases, etc, we may see computational chemistry start to predict novel materials (via machine learning, for example).