Every editor is different, including vim. In Sublime Text 3, it's Shift+Delete to delete a line, and whether it goes to your clipboard or not depends on a config setting. There's a shortcut to duplicate a line too, but I don't remember the default because I've customized it.
Ctrl + X is the shortcut for cut (as in cut then paste). It cuts the current selection. It doesn't auto select the line if there is no selection.