It looks like I have found the cause of the issues

After each fresh install, I directly configured my own template. Today I didn't, and it worked, so I replaced the default template with my own template, file by file, and then found the cause, it was/is my template :|
The error was made in "startpage.pstpl", I placed (copy-paste) a php tag in the template, which was (logically) displayed as following:
<link rel="stylesheet" type="text/css" media="all" href="<?php bloginfo( 'stylesheet_url' ); ?>" />
Why this caused the first question(group) no to be saved the first time, is still unclear to me. I have tried it with multiple browsers, and all didn't save the first question(group) the first time. After removing this line, the entire survey just behaves like you expect it be behave.