"dNg" is the Adobe "digital negative" extension.

The Firefox download should have a "dMg" extension (application package).

If you rename the extension of the downloaded file, does that work?

Also, are you sure you got it from Mozilla directly? Just in case you don't trust that one, you can try again from here (scroll down to your preferred language):

https://www.mozilla.org/firefox/all/

I thought I downloaded it from the Firefox site, but I did it again from the link you gave me. The download does say .dmg, not .dng, but the same thing happens! PhotoMechanic opens!. I tried to attach the download file so you could see what happens, but it is grayed out.

Okay, that's strange. Could you try this: either right-click or Ctrl+click the file, and see whether "Open With" will let you use Disk Utilities or Disk Image Mounter.

You also could try Apple's help on fixing mistaken file associations. Not sure if this applies to your version:

macOS Sierra: Choose an app to open a file

That got me the little fox icon, but the program won't open. It says my profile is missing, but it doesn't tell me how to create a new one.

Did you have Firefox installed before and clear out some files??

You can create a new profile. Please see this article: How to run Firefox when your profile is missing or inaccessible. If you don't get instructions for Mac, you can customize the article using the "Editing Tools" section in the left column of the article. There is a drop-down to select MacOS.