Finding keys for anything is like that on GOG.
I've just wasted 15 minutes trying to find my key in GOG Galaxy.
It's only available through a web browser, in your games library.
There's no need to do this anymore if you want to access the donator features. This was a bug and has been fixed in the newest version.
No, you should still register simply because you have the keys to register. GOG versions will sometimes come out slower than on-site (they don't always but it happens), if you're a mac user specifically the on-site version updates way quicker/more consistently and dealing with DLC/addons is a bit better. There has also been a report or two about GOG-specific bugs in the OSX version.
I don't really remember about how the item vault works in GOG anymore, but I remember my stuff still going to my ToME account so registering your stuff may be needed for more vault space? I need some confirmation on that.