To what extent do parents have authority over children? Can they choose a child's career? Can they choose where they are allowed to live? Can they choose whether or not they can go on a spiritual retreat? Say a child wishes to forsake the world for 40 days, go to the mountains, and start doing duas and prayers for 40 days, can a parent stop the child from doing such a thing?

If he is Balegh, parents has no authority over him in all what you mentioned. However, regarding traveling, if this hurts them, he should not do it, as hurting them is not permissible.