Your CD key is on your account in the GOG website. Look through your games on the website, find the original DOOM 3 and from the drop down menu you should see your a Serial key option where you can get your CD keys for the base game and ROE.

Thank you for replying,and your help.It is weird because with the Steam version i had no Problem like this one.

Actually the only game so far with a CD key, not sure the reason. However, it is basically an non altered original version, and CD key was part of this version.

I just hope there will be no issues with further installs, but as long as there is no "online check" i basically can activate it as many time as i enjoy.

It has to be done in your games library under your account name IN YOUR WEB BROWSER! NOT in the GOG App. Which is something no one is telling you here. Then from the top account name at the top of the page, NOT the games choice on the page but the games choice in the pull down menu that pops up under your account name at the very top of the page. Then choose games. You'll know it's correct because the next page says MY Collection. And shows all your games. Choose the non BFG edition of Doom 3. Click more next to System and Language. The second choice in that menu is serial keys. Doom 3 and the RoE expansion. You need both. And they can't be copied in the web browser so you'll need to write them down.

I think they just have forgotten to put the keys into base\doomkey and base\xpkey.

Weird, I was able to start the original game just fine. Then again, I recommend people use dhwem to play the classic Doom3 as it offers widescreen and can be used in Linux. Still, the vanilla installer should have worked, unless the install went wrong...