Many clips in my browser/timeline have a green label over their names as opposed to the standard light blue, or none at all. I'm not having any trouble with these clips (yet) but I'm worried they might pose a problem at some later point.
What are these labels? Are they cause for concern?
Those labels represent that you need to do a "preview" render. The clips or graphics that have this line might not appear as sharp or crisp in FCP as you would expect, by rendering the preview, the clips or graphics would appear clean looking.
They are not a cause for concern, you can keep editing with the lines there and your timeline should playback fine. When you export, FCP will render them for you I believe, which may increase the time it takes to export.
In short though, no they are not a cause for concern.