I can confirm that it is most likely a DNS problem.
to fix it:

• set DNS to 1.1.1.1 (2nd:8.8.8.8)
• in command prompt: ipconfig /flushdns
• close downloards FAF client
• actually close the application completely via the task manager (it did not start; the process was still running. Not sure if it kept running from first launch or did not start completely from second start)
• start downloards FAF client

result: logged in correctly