Hi,
First, on losing emails -- if those are IMAP accounts, your messages, including old ones, should have stayed on their respective mail servers (and the app on the new device should have picked them up).
Second, on the app not receiving mail --
Please check the device's system settings -> data usage -> Menu -> make sure that "Auto-sync data" is enabled.
This is a device's wide "kill switch" for all background sync, and Aqua respects it being off. The setting default to "enabled", but apparently, sometimes it gets switched off by all too helpful sales staff / techs who do initial device configuration when you buy one (just a theory, but...)
Third, does the app receive mail if you use the refresh / sync buttons in the app's UI (as opposed to automatic background sync)?
Finally, feel free to contact me at kmansoft / gmail, so we can have a more expedient discussion.