I was typing more detail, when I suddenly found the solution.
At the Mail section, after the 'From' input, there only stood '[your-name]'. Now I have added '<[your-email]>' and the email problem is solved.
One question left: I've sent a test message with a Gmail email address. When I open that mail Google says the following error message: https://support.google.com/mail/troubleshooter/2411000?p=sent_warning&rd=1
How to fix THAT problem?