I recently had a major hard-drive failure need to access the images from my back-up drive. The file type on the back-up drive is .nef but my catalog files and the images I lost were .dng. So now when I go to my catalog and I'm searching for the missing files and I go to my back-up drive, because the files there are .nef the catalog doesn't recognize them, of course. So what do I do? And in the future, how do I save my back-up files as .dng?
You can download the free Adobe DNG Converter and convert your NEF to DNG, which Lightroom will then recognize.
In the future, you want to make backups after you convert to DNG, not before.
The extra copy made during import is indeed equal to your original from your camera, in your case thus .nef.
I make every time as Lightroom advised to make a backup of the catalog, usually once a week, een full copy of all the fotos in the catalog. This is an exact copy including the structure of the Lightroommaps as well.