<p>In our Flex AIR app, we have the problem that our main app window is fairly narrow. This means Alert dialog boxes are chopped on both side, while the right click menu is cropped. How can we get these windows to not get cropped by our main window?</p>

With flex applications running inside of the flash player, windows cannot be shown outside of the stage.  So the only way to make this work would be to make your app larger.
You could though use the flex/ajax bridge and call a javascript alert box instead, they would not be bound by the stage.  But it would not be skinned like the rest of the application though and would take some more work to get hooked in, especially if you are listening for the user to click the okay button...