No, multiple classes do not work via KindleGen (unless this is one of the bugfixes for 1.2), and as you've noticed, it also completely ignores the first class used, which I think technically it should try to render under CSS fallback rules.
You also can't do descendant selectors, even the generic catchall type supported by CSS1, so if you want to style, say, all list items within a div of a certain class type (or even any div in general), you're going to have to mark up all those <li> or at least their containing <ol>/<ul> separately with target classes unless you're willing to have the style you want apply to everything within the parent <div> to save effort.
Those are just two of the many ways in which KindleGen in particular and the Mobi format in general are made of fail, lose, and frustration. I say this from bitter personal experience.
Hope this helps!