Did some more digging and found that, for the most part, this is SOP and that it's working the way it should. My incoming iCloud mail hits my desktop and iPhone inbox simultaneously. The POP fetches every 15 minutes. I had thought that POP was behaving the same as iCloud, but in retrospect, it was just coincidence that both accounts delivered at the same time in some cases. Ah well, the learning curve continues.

In the meantime, I've installed mobile banking apps for all my accounts, QR Code Reader, DropBox (wifi only enabled for now), GasBuddy, LinkedIn, USPS, and the new Google Maps.