When you right-click on at item on the Start menu, no shortcut menu appears, even though it is supposed to. You want the shortcut menu to start appearing again.
Using a graphical user interface
When you right-click on an item in the Start menu, the shortcut menu that is supposed to appear gives you a variety of options, including running the program or opening the file, removing it from the Start menu, copying or moving it to a folder, and so on. The exact menu will vary according to whether the item is a file or a folder.
There's no logical reason that turning off dragging and dropping should also turn off the shortcut menu for items on the Start menu, but it does. When dragging and dropping is turned off, you won't be able to drag items from one part of the Start menu to another and as you can see in this recipe, it also kills the shortcut menu for Start items.
Sometimes the shortcuts and folders on the Start and Favorites menus may not be alphabetized, or new shortcuts may be added to the bottom of the menu. To fix the problem, see MS KB 177482, "Start Menu and Favorites Menu Are Not Listed in Alphabetical Order."