I am busy with my header and footer.
When I put this into the header in my head tag, it destroys the template.
The script already includes jQuery so if that's still in place including it again would cause problems - that might be all it is (included by line 28 in the distribution html/header.php)
<script src='<?php print $config_baseHREF; ?>html/vendor/jquery.min.js'></script>
I removed it but the link takes care that the search is disappeared.
As the distribution template is based on the Foundation framework including Bootstrap CSS etc. is likely to cause problems as they are different approaches to achieving similar functionality.
Normally, style definitions applied last take priority so you might have some success by making sure that your custom includes come first within the header with the Foundation links the last CSS links in order...
it works but the searchform is completely gone.
Everything within searchform.php is gone.
I assume this causes the problem:
When I change the class, the searchbox appears again.
Can I change this without any problem?
Sure - if everything is functional that won't cause any problems that would not be immediately apparent.