It's not OS issue. Using Linux will not resolve OP's problem... he must disable auto updates...
I don't think so. I use Linux Mint and have the OS to only update manually and it will offer Firefox updates but I can install them whenever I want. I do not think it can do it without me inputting the password.
Of course it can't, unless the application was running as root. If you use Linux and you use root as your user account - or set up a user account with equivalent rights - you just deserve what you get.
Another possibility would be if the application was installed not "system-wide" but locally. That would be relatively uncommon, especially for an application that has a distro package on all distributions that I've ever run into.
For people with such a setup, I'd be curious to know the details, please.
Note that on Windows, Firefox (and Thunderbird) resort to an awful trick to allow self-updating (whether automatically or not) *without asking for an admin password*.
They install a service (by default) that basically allows bypassing UAC for the Mozilla products. How sweet.
So If you have ever wondered why Firefox was one of the very rare apps that never asked for a password for updating, now you know.
This service can be disabled of course, but you have to go to the administrations tools and manually disable it.
How sweet from Mozilla, they probably wanted to "lower any friction" to the use of their web browser, as they say in marketing.
I'm personally a long time user of Firefox because it just works for me and is still one of the best for privacy. Also I want to encourage alternatives to the Chrome monopoly.
But the attitude of the Mozilla foundation, unfortunately, I find more and more unacceptable.