I mostly agree with you. Just one point I disagree with is the feeling of emptiness. A symptom of Depression itself is feeling empty and/or numb (source: Depression: Recognizing the Emotional Symptoms). I feel empty when I am deeply depressed. Feeling empty or numb constantly is a different matter though, so feeling constantly empty or numb regardless of state is more a sign of a Personality Disorder than feeling empty or numb during a depressive episode is. Also remember that some medications can cause a feeling of numbness, so that may be as an effect of the meds rather than the disorder itself.

Your definition of a mixed episode is the correct one. It's a mix between symptoms of mania and depression which exist concurrently. Hence its as straightforward as it sounds; it's exactly what it says on the tin - a mixture of manic and depressive symptoms. This doesn't mean cycling between two different states.

As for rapid cycling, it is not a diagnosis in itself. You cannot be diagnosed with "Rapid Cycling Bipolar Disorder". Rapid Cycling is an episode descriptor, or more accurately a "course specifier" and isn't a permanent thing, in the vast majority of cases. It isn't a mixed episode, or even similar to one. Only around 10-20% of Bipolar patients are thought to experience a rapid cycling state at some point in their lives.

I do think you're right about having emotions during episodes, like anybody else would during eurthymia. We all have good days and bad days, or good hours and bad hours in fact, so I don't see why that would be different during a depressive and/or manic episode.