Is breathing through the mouth forbidden and contrary to God's commands? Is a sick person permitted to do so if their nose is blocked? Is what doctors are unable to cure considered envy or sorcery? What is the ruling on going to a psychiatrist?

There is no harm in breathing through the mouth when the nose is blocked. The default is that it is permissible even if one can breathe through the nose, but doctors prefer breathing through the nose for air purification. There is no harm in visiting a psychiatrist or any other doctor. The inability of some doctors to cure a disease does not mean it is magic, for Allah has sent down a cure for every ailment except old age. There is no harm in performing Ruqyah (definition) (Islamic exorcism) for magic or other ailments, even if the person is not afflicted.