How do I move a docked control around (or change the dock order) in my Form?

You cannot move it via drag-and-drop. But you change it’s position or dock order by selecting the ‘Bring To Front’ or ‘Send To Back’ verbs in it’s context menu. ‘Bring To Front’ will move it to the top of the children list and ‘Send To Back’ will move it to the last of the children list. This is possible because the docking logic will be applied on the children in the order in which they appear in the child controls list.