Good news
and bad news
Good news first: As the filtering clear and also the saving was not working in the frontend and also my other cook projects were not saving any longer in the frontend and ACL checking did not reveal any permission problems, I decided to install on a pristine Joomla 1.7.3 installation (with sample files) and voilà: Everything works. I can reset all my 4 filters with the "clear" button both in front- end backend and I can save as usual.
The only small incident: When installing the component I get the following notices for every filter:
Notice: Undefined property: JDomHtmlFormInputSelectCombo::$href in D:\Wamp\www\17\joomla\administrator\components\com_fursongsheets\dom\html.php on line 162
Notice: Undefined property: JDomHtmlToolbarDefault::$target in D:\Wamp\www\17\joomla\administrator\components\com_fursongsheets\dom\html.php on line 165
Now the bad news: Although I never changed the default ACLs in my testing environment, I cannot save any longer in the frontend with all my 3 Cook-projects. As my standard testing environment has many other extensions installed, there must be some interference with some other extension. I will try to figure it out as much as possible by deinstalling some of the recently installed 3rd party extensions.
So setting and resetting multiple filters (mix of foreign key and enumeration fields) is now working! Many thanks!