If you disable all the DLC except Aztec Civilization on your iMac before starting a game that you intend to play on both devices, you should be able to start a game on the iMac and open saves from it on your iPad. (I can verify this works on a Mac Mini and don't see why it wouldn't on an iMac.)
You can! I've been switching back and forth from my iPad Mini 4 since I got the app. Use the File Sharing pane of the iPad settings in iTunes. Here is a screenshot:
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