For example, if I were to write a calendar app on top of AIR, say with Flex, could this app pop up reminder windows for approaching appointments, just like Microsoft Outlook can?
Clarification: Can those windows be actual dialogs where I can enter and save information?
See Creating toast-style windows
Twhirl pops up "toast" notifications (similiar to most instant messengers), while it is running in the system tray.
So yes.
There are notification balloons tips but this method is going out of favor with systray icons in Windows 7 and it's not cross platform. Unfortunately, you can only call this in the Win32 API using Shell_NotifyIcon and there is no way to get to it from Air.
Stuck making your own toaster popups.
YES
Your Flex AIR application can pop up windows, dialogs, toasters, whatever.
Have a look at this article on the Adobe Website, for creating your own Toast Style Popups.
(Credit for this goes to Duncan Smart, answering this question.)