OK I solved it:
1. I changed the database prefix during the site setup, but I did not change it in the user role tables. Here is the fix: https://wordpress.org/support/topic/change-user-roles?replies=15#post-1970662
2. User role settings seemed to be broken at all, because 1. did not fix it completely. So I wanted to edit user roles anyway, installed the module "User Role Editor" https://wordpress.org/plugins/user-role-editor/ and there was the option to reset user permissions to wp-core settings. I did this and everything was fine.
3. The solution for
PHP Notice: Undefined index: action in /wordpress/wp-content/plugins/wordpress-mu-domain-mapping/domain_mapping.php on line 632
was described here: https://wordpress.org/support/topic/notice-undefined-index-action-2?replies=2
Afterwards, I performed successfully an update to WP 4.4.1.
