The source code zip is the right download.
You can then upload all the files inside the source code zip to a web host. Some examples include our partners Pebblehost (which is a full web host), Named Hosting which is specifically for Nameless, and NamelessHosting which handles all the files for you.
It looks like your domain is set up as a redirect to an IP address.
The best thing to do is modify the domain so it no longer redirects (such as using an A record) - your web host can help you with this.
If you do this, and you still experience the issue, please let us know.
Exactly, as you say it looks like Contabo is routing the request through its own systems.
Can you please open a support ticket with your web host, they might be able to help with the issue.
Looks like that website is using XenForo, but this Nameless template looks similar: https://namelessmc.com/resources/resource/142-modernx--premium-template/
To get more information about this error, you can edit the file core/classes/Email.php in a text editor. Delete all code inside, and replace with this new code.
Then, when you send a test email in the StaffCP, a full debug log will appear on screen with the exact communications between your web server and the SMTP server. This should include some hints as to what the problem is.
Once you have the full log, revert the contents of core/classes/Email.php back to its original code.