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Manchis7: Doesn't works...
James_Neave: I found a solution to this here:

Under "Settlers® 3: Ultimate Collection - Game asks for CD-Key in singleplayer mode"

If you are using GOG GALAXY, select the game and click the customization icon (next to PLAY):
→Manage installation

This fixed the CD key issue for me after I had fixed the not-starting issue by using the offline installer
How do you download the oofline version?