OK, now that makes sense. I've left it at the default of 15 minutes, but that still doesn't explain the hourlong delays. I have the one Exchange account, and two IMAP accounts. Well, I guess so long as it syncs eventually that's all that matters. Calendar and Contacts can wait, but email has to be instantaneous.
Speaking of push, for my two GMAIL accounts I use the native GMAIL app in my phone, which pushes their messages in real time. If I set up my Gmail accounts in AM, can I use the push feature that way, or will that only work with the native app?
One last (unrelated) question: Where is the setting to have AM open onto the main home screen every time, regardless where it was when I closed it? Whenever I open AM I'd like to see all 3 of my accounts with their unread totals.