I'm beginning to wonder is there is some security software that is somehow blocking offline data somehow. Bare with me, there is some investigation questions below. You seem to have some pretty good technical abilities so I'm going to give you some advanced hints as to what to look at.
When you are in FF and online, do you then see both the offline and online missions? Your online screen should look the first image below.
After you are clearing all the caches, when you are first downloading a mission FF will open a message at the top of the page (its not real obvious so its easy to miss), this message will ask you if the site can store data on your computer, see the message boxed in red on the 2nd image below. It's possible that you won't see this message depending on the setting "[] Tell me when a web site asks to store data for offline use" in the FF Options box. (see the last attached image below).
If you are not seeing offline missions when you are online like the image below, then there is a problem with localStorage. localStorage is different from the offline page cache and its where MM stores all its offline mission data. There is a FF addon that lets you view this data here:
https://addons.mozilla.org/en-US/firefox/addon/firestorage/. If you see the missions when online, but they are not there offline, I'm at a loss!
Good Luck,
Mike