"Why would I have everything rise and fall like that one after another in sequence-and even the Nitrates falling to almost zero...but the ammonia still be a bit high like that?"

Two possibilities here.
1. The ammonia test kit could be wrong. Test kits have a shelf life. There should be an expiration date somewhere on the box. If you can't find the expiration date or if the box is still technically not expired but you doubt it, you could buy a second test kit of a different brand and test your levels with both test kits. It may be that one test kit is reading 1 ppm and then you double check it and the second test kit reads 0 ppm.
2. Nitrates don't just fall to zero without a water change or being eaten by plants. As long as you continue to add food to the tank (which you should never stop doing, even when there aren't fish in the tank, or it will uncycle) there should be a steady production of nitrate. If nitrate is dropping that means that the nitrogen is getting clogged somewhere in the pathway and is accumulating as either ammonia or nitrite. My bet is that this is what's happening. The tank is not fully cycled until ammonia and nitrite are 0 ppm and nitrate is steadily increasing.