During class selection, for some reason, the classes are all mixed up. I click on "Rogue" and get "Healer". They're all mixed up this way.

This is on a fresh download from Gog. Tried with all mods off, and the same thing.

Is this just a fubarred download?

I think you're referring to the Daggerfall GOG Cut (which is outdated and not supported). If yes, do yourself the favour to install Daggerfall, and then download and install Daggerfall Unity (the most recent version). Should you want to play mods, too, you can always get them from Nexus (the ones included in the DF GOG Cut are listed on the store page, and there are many more, and likely more interesting ones - but this is a matter of personal preference).

With that said, the problem you describe shouldn't exist even for the outdated version of DFU. It might be caused by one of the mods (or a conflict between mods), or by missing or corrupted game files. But this is just a guess - I don't play the GOG Cut.

Well, I was referring to the Gog final cut, but I tried downloading the 1.0.0 release and it's doing the same thing.

Doubleclick on "Barbarian" and I get Mages. Etc.

There is no "final" cut.
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Did you play DFU 1.0.0 standalone, or did you replace the DFU version of the GOG Cut with DFU 1.0.0? The latter may not work properly because many (or all) mods are outdated, too. It's also possible that there is something wrong with your Daggerfall installation. I just tested DFU 1.0.0, and choosing a Rogue or Barbarian works (which province did you select?).

If you haven't done it already, install DFU 1.0.0 standalone AND reinstall Daggerfall, point DFU to the new DF installation, and start a new game (without mods). If you get the same problem again, you could try deleting the DFU folder in AppData\LocalLow\Daggerfall Workshop (it contains config, mod and savegame information, and possibly leftovers from DFU GOG Cut - make a backup before deleting, if you want to keep the files). Then try again.

If this doesn't work, too, please describe EXACTLY what you do and what you see: You go to class selction, double-click Rogue, and then what happens? You ought to see a confirmation that you've selected Rogue. Is this where you notice that the class is wrong?
PS: I don't use Galaxy. If you do, please consider trying to install Daggerfall standalone (just to rule out the possibility that Galaxy might play a role).