The Duplicate operation makes copies of all the selected objects. The copies will be given the same name with (copy), (copy 1), (copy 2), ... as suffixes. This is the exact same operation as the Duplicate button in the Object Browser. The hotkey for the Duplicate command is Shift-c.
There is not a specific menu item in the Edit tab for the Discard operation, which deletes all the selected scene objects (Meshes, Pivots, etc). However you can use the Delete key or the x hotkey to Discard.