UPDATE: The error "400 .... Use PUT to insert or update list members" appears only when "Use double opt-in" is selected.
However the message "Thank you! Please check your email and confirm the newsletter subscription." comes if "Use double opt-in" is turned on or off. (When it is turned off, no email is sent.)
I saw in the release notes that this error should have been addressed.
I have no idea how to handle this issue. I checked the forum, reinstalled on joomla and also started from scratch again on mailchimp.
Any help on this would be appreciated.

Another UPDATE: I spent many hours on this but can not get it to work correctly.
What I now saw is that the script for CMC is loaded twice. Could that be the problem?

<script>

jQuery(window).load(function(){jQuery('img.caption').JYAMLCaption();});

jQuery(document).ready(function() {

    new cmc("#cmc-signup-115");

});

jQuery(function($){ $(".hasTooltip").tooltip({"html": true,"container": "body"}); });

jQuery(document).ready(function() {

    new cmc("#cmc-signup-115");

});

	</script>

I disabled jquery in the module. I'm completely lost on that.
Here's the link to the test website: http://aef.debonis.ch/