The first methods seem to work in the browsers that I tested, but the option tags doesn’t really correspond to actual elements in all browsers, so the result may vary.
Just use the selectedIndex property of the DOM element:
alert($("#dropDownMenuKategorie")[0].selectedIndex);
Update:
Since version 1.6 jQuery has the prop method that can be used to read properties:
alert($("#dropDownMenuKategorie").prop('selectedIndex'));