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.

Finding keys for anything is like that on GOG. 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.

Mostly, you should register because the ingame achivement are linked to your TOME account and completing them is necessary to unlock classes and races, which for the majority are locked at game start. This mean that if you play in another PC everything would be locked again unless you login with the TOME account ingame.
Furthermore, I suggest registering the various DLCs too. This way you gain more donators levels, increasing the storage slots in the online vault.

On the other hand you can copy the unlock file between different computers and stay completely offline.

I've registered my copy at te4.org some time ago to circumvent GOG's slow DLC updates, but I always play offline.