Perhaps I'm supposed to click on the fairy avatar and then paste there? I think I tried that and nothing. It was as if the paste buffer was empty.
No, you should been adding wings to the timeline, go to animation track and copy that. Suppose that is what you've been doing....
Try to replicate the clip by dragging it to the right with Ctrl+Left Mouse CLick.
Or select a Loop icon in the timeline menu, hover mouse to the end of the clip until a cursor changes to the plus with arrow and then Left Click and drag the clip to the right to replicate as many times as you wish.