@catzilla I took a look at Hostgator, and it does look like they offer both FTP and SSH/Shell access. Here’s a video from them about how to enable shell access, and here’s their documentation. Here’s their page on FTP.
Shell access is the command line – it’s kind of like a MU style in that it navigates through your website files via typing into a terminal. FTP is basically file transfer – you can download copies of files from your website, or upload files to it. I’ve used Termius lately for a project, especially because it has both the command line terminal and FTP in one. When I want to edit LocalSettings or something like that, I usually use a code editor (BBEdit in my case, although I think it’s Mac only); I think it’s common for code editors to have an option to open files directly from a server, so I find that easier than trying to edit thing on the command line.
For CSS editing, yes, it’s pretty much normal CSS. You won’t want to paste in the whole code from another skin here, that won’t really work right. If you found a skin you like, what you’ll need to do is download the files and upload them into your MediaWiki’s /skins/ directory; I usually use FTP for this. And then on the skin’s page, it usually has the line of code you need to add to LocalSettings.php in order to activate the skin. Then you’ll either need to add a line of code to set the new skin as the default skin for the wiki, or go into your user preferences on the wiki and just change what skin you’re using. (You can also deactivate other skins in LocalSettings.php so they can’t be used.)
I can’t see what CSS you tried to add to the Common.css page to know why it might have updated one link and not others. I can say that I usually have to do a hard refresh on a page after editing one of the in-wiki CSS files to make things show up properly.