command V not working on Mac Air

Since you can paste from the menubar but not using a keyboard shortcut the problem is 1 of 2 things. 1) either the command or V key doesn't work - but you'd certainly have already noticed that or 2) a program is grabbing that shortcut for its own purpose. Open System Preferences, select Keyboard, click on the Shortcuts tab and then in the window pane to the left click on App Shortcuts. Do you see Command V being used by any of the shortcuts listed in the right window pane? If so, select that shortcut and either changes its shortcut key by clicking in the shortcut box and typing your own key combination or clicking on the minus sign below the box.

The other possibility is that you have another program that creates keyboard shortcuts and snippets.