I'm new to Final Cut Pro X and I'm trying to extend the duration of a static PNG image so that it shares the length of the video that's over it, but it won't extend no matter what I do. I've tried Control+D and typing in a duration but it just does nothing.
At the end of your main timeline add a gap Option-W. Drag and drop the still picture behind the gap into the main timeline. Change the duration as needed. Move the clip where you want it.
Your behaviour you described isn’t normal, it sounds like you do everything correct. Above is to eliminate any timeline issues from stacking and try to sort out the problem.