The software Dropbox provides an shell extension which adds context menu items to all files in a specific folder. One of these generates a public link to view the selected file.
In a C# tool I want to call this entry without any user interaction. I want to achieve the same behavior as if the user clicked on the context menu item of a selected file.
I know that the shell extension is provided by a DLL, is it possible to make a call to this DLL to achieve the expected behavior?
Shell extensions implement IContextMenu and it is possible to execute menu commands without showing a menu (See this blog post for details about "hosting" IContextMenu)
Once you have the menu, you would call IContextMenu::GetCommandString and look for a specific verb, if Dropbox does not have a somewhat unique verb, you are going to have to do something hacky, either match by menu text alone, or call the Dropbox shell extension dll directly (DllGetClassObject export) and fake everything (Pretend to be COM and shell) or if you know the CLSID, you can at least get help from COM and just do the shell part.
There is a freeware tool called runmenu that allows you to play with shell menus/IContextMenu (I'm sure you can find a copy somewhere)