Try to go to the browser menu, look for "help" or "about firefox" and probably there is a way to update itself.
If it wont work, you can download any other browser, I believe Palemoon is XP and Vista compatible and the website may work.

https://www.palemoon.org/download.shtml

I use the Portable version wich don't need install, just download and run.

Also, keep in mind most GOG games are optimized to newer Windows versions (like 7, 8 or 10)

Hey i always used backups and had like an external HDD since 2008 also when i got my first internet on a usb stick, now have 3 external HDD, don't think i ever lost any important data.
Actually i was in middle school when i bought my pc, job maybe brought me some money but not for long since i needed to pay bills on when i lost my job. Couldn't afford to buy a second pc. Now payed some tax and now there should be some money left if the bills won't stop rising.

I had an external once but it was a crap one and it died and I got soured on externals for a bit. I do try to backup stuff now, though.

Over here many americans sadly cannot get jobs until later on(child labor laws), or at least ones that pay good enough, so I couldn't get one without parental help until I could someday get one on my own.

Which brand of external HDD did you buy that it died?
I always used western digital and they still work now.

Dunno, it was over 5 years back....and it was all black with rounded corners in the front.

I mainly wanted to know if you get a zip file that is all you need or an exe installer that requires internet connection when run to install. If the former, I'm all set. If an installer, I could download it, transfer it and run it, but there was a couple of ways that went wrong when trying it for alternate browsers. Some gave me 'Not a valid Win32 program. Another ran but required service pack two. A very longstanding issue with it that never was figured out left it unable to install service pack one.

If it's an exe installer, it'd be nice to know that being unable to successfully install would be grounds to get a refund. I wasn't really asking technical questions, but rather pragmatic ones. I've found workarounds for the meantime. I just wondered how what you buy comes so I could jump on the sale.

You appear to be talking about several different things here all overlapping.
First, windows vista is old, out of support (and a terrible operating system). You should change this as soon as possible regardless of anything else you do.
Second your browser issues, most likely caused by using an old outdated vista install. Things move on and failing to keep up will cause compatibility issues.
Third, downloading games. Yes, you can certainly download offline installers, they are in your account, go to the website with browser, click on your user name and select a game. Under offline installers are the files you need to download, download them all, and take that time to back them up as well.
Fourth, if you have game specific issues, such as not running because a service pack - which operating system specific - then you would need to fix those. Again, a lot of that will be because you are using an old, unsupported operating system in vista. Refusing to update it will cause you issues and may mean you can’t get refunds. Check the game specific page on the website for specific supported operating systems, some are 64bit only so won’t run on your os for instance.

So, to reiterate, to fix 90% of the issues you experience, ditch windows vista at the earliest opportunity. Move to windows 7, or preferably windows 10. You will then be able to run mostly everything out of the box, including modern browsers. Almost all new computers come with win 10 preinstalled, or you can pick up an oem version real cheap.