Here are some suggestions of mine...
https://fsnebula.org/workshop/mods Looking at this...
That page needs some real prettying up. Mod ID I think could be hidden and just be for back end internal use. We've got Mod Title anyway. Let's have the mod bitmap be shown, and perhaps the beginning of the mod description (first 140 characters or something?). Maybe authors could also upload images and/or a link to a youtube video that should show up (as links) with the visible description? First release is nice, but also a "last updated" column, as well as the download counter there too. Also filters! I saw you added those categories before, but I don't see them being used for organizational purposes yet. I think it would really help if there was like a FSO Builds category. It's not obvious to a newbie what they need to actually play.
So there'd be a tab/screen of FreeSpace Open builds, which would all be clearly marked (FSO alone doesn't mean much to a newbie and not all mods guarantee compatibility with the latest and some might need custom builds), and then tab/screens for the different categories. Also I think the MediaVPs should have an unfair advantage with placement so its like a no-brainer to select. So a newbie would be like "ok i have this installer, oh builds, right that's why I'm here. I'll take the version with the biggest number and I'll take these MediaVPs which is labelled as super recommended and then I'll explore these other sections"
I'd also like... just recommend most of those changes to the installer as well. I think ideally both should act as possible in terms of functionality and style.
Launcher specific... Maybe there could be something like a "FSO exe ready/missing/lol what are you doing this is still retail" sort of indicator? Someone running this for the first time may not have properly selected the right exe yet. Like in wxLauncher, it clearly tells you which build you are using. Being quickly able to change builds from the main window is super useful for us devs, I'm not so sure about normal players though... Maybe an advanced mode?

(Also the launcher can't find FRED right now

)
Sort of related as well, wxLauncher can take a build filename and basically tell you what this build exactly is. Like fs2_open_3_7_5_d is FreeSpace Open 3.7.5 (Debug) which could help people identify special builds. And nightlies have githashes (useful for only them) and a build date (useful for everyone else) so having that info clearly readable would be cool.

Also I think Settings needs to be broken up a bit. Like right now. "Settings" at first I thought it was just launcher settings, and didn't have anything to do with FS Open. So maybe a button that takes you to Launcher Settings and another button that takes you to Game Settings (which can still be the same window in the end, just different parts of that view).
I don't think I'd call the Default flags what it is right now, maybe more like "advanced default settings"?
And I see you have a button to the fs2_open.log which is amazing and I had just spent a paragraph saying you should add so that's awesome. But maybe also as shortcut to the proper config folder (like the right %AppData% folder on windows, or %OSX_EQUIVALENT% or %LINUX_EQUIVALENT% if we get that far)? Gotta be easy to find those screenshots!
Also re: your points...
Does the auto-updater on Windows work?
It didn't for me... I opened it, it didn't say there were updates automatically. I just downloaded a new exe and grabbed the new version.
Is anyone using the fso:// links? (Mod pages on fsnebula.org have a green "Install" button which, when clicked, should launch Knossos to install that mod.)
It did for me!
Should I display the forum highlights? Where?
I think you need a welcome tab. Grab the highlights board rss feed or something and mostly populate it with that. And maybe have a link to whatever welcome guide that the main site will end up having?