What does downloading the game with GOG Galaxy (as opposed to getting the game files) do? Is that really necessary for multiplayer? I'm not an expert on what multiplayer options Stellaris has. Do any of them not work with the offline install? Does the game have a "check for updates" button that allows you to get updates from online and have them installed in a very straightforward fashion without Galaxy?

My old computer seemed to run slower when I ran both GOG Galaxy and Steam at the same time (I have different games on each; I did not run multiple games at once, but might close one and open another without logging out of the service). Hopefully that wouldn't be an issue with my new computer, with a better processor and an SSD, although I'm on the same internet connection.

Yes it is... or you can use your Paradox key to unlock the Paradox version (linked account needed) & use that for multiplayer instead no, you get last 5 builds with Galaxy... a few special builds with Paradox and bugger all for Mac or Linux not something I would do & nothing to do with how good or bad your system is... they just don't play well together

the current state of muliplayer is;
Gog (Galaxy) vs Gog, Paradox or Steam (test server)... all players must have the same build but different dlc is fine