How can I properly set the triggers/storage qualifiers in Signal Tap II so the data I see after it is run is from the initial trigger to the end of the sample depth that I specify?

I'm not sure what you are referring to by the "start condition" unless you are referring to the start/stop storage qualification. That is for just qualifying what data to store in the buffer, not to actually trigger, which may be why you are not getting the logic analyzer to trigger on its own. You can specify multiple trigger conditions and additional columns will appear in the Node List. With a standard sequential trigger, when the final trigger condition column occurs, the logic analyzer stops and captures data. You shouldn't have to manually stop the logic analyzer if you've set up the final trigger condition column to match what you're looking for. Perhaps posting a screenshot of your logic analyzer setup would help here.