What an adventure! (I'm not complaining, I knew it was gonna be.) But I think I'm gonna write a post on this, because there's a LOT that has happened. I downloaded firefox 55, unpacked it, and whenever I launch it from that folder, it works like a charm. However, it is not the browser installed into the system, so any attempt to access the browser in any other way gets me the out-of-the-box firefox 44. I managed to remove it once, but it won't let me install the 55 one, just the 44 back again.

I tried it with the GDebi Installer (and the appropriate GDebi package I downloaded), I tried it with the Synaptic Manager, even with the Terminal commands. For some reason it won't let me. Meanwhile various websites tell me to update my browser, or flat out refuse to give access (such as Slack). Oh yeah, I also tried to use the Software Updater, but according to that, I'm already using the most recent version of firefox (that is 44). 
Like I said, no complaints here, and I knew Linux was not gonna be easy, but this is a bit weird... Anyhow, I'll keep trying around, and accumulate material for a post.