Hi,
In the timeline, if you do a Modifiy/Split clip and then click on one of the splits to select it (it will get a yellow box around it), and then do an Edit/Copy, Edit/Paste, it should work to paste the selection in the timeline. Same with doing that to a full unsplit clip.
Also, you can hold down the "r" key and press-drag your cursor over a portion of a clip and do the Edit/Copy, Edit/Paste. Sometimes you may need to move the playhead to a blank space at the end of the timeline before doing the Edit/jPaste.
If the above are not working, try opening iMovie while holding down the Option and Command keys, and selecting to delete preferences in the box that appears. Sometimes that cures glitches.
You might also try doing an Edit/Select All, Edit/Copy, of all the media in your project and Edit/Paste it into a newly created project. See if then you can do your Copy-Pasting in the new project.
-- Rich