[client]: Failed to copy AIR descriptor to the output folder: C:\Users\Chelsea\Documents\IntelliJProjects\out\production\client\Client-app.xml (The requested operation cannot be performed on a file with a user-mapped section open)

First of all check that the path to the AIR descritor template configured in IntelliJ IDEA is not equal to the path that you see in the error message. You should keep template somewhere in the module content root or source folder, but not in the output folder.
Second, find some unlocker utility in the internet and see what program is keeping a lock on the file. Does this file exist when you see the error? Are you able to edit or delete it using any file manager?
Try to disable 'Use external build' at Settings | Compiler. May be it will work better this way.

I guess is that I have an Empty project with a pure actionscript Flex module inside (as opposed to a Flex project with Flex modules)

IntelliJ IDEA project doesn't have any type. Only modules have types. To work with Flex or ActionScript you use Flash modules. Each Flash module contains one or more Flash build configurations (BCs). Flash BC may be pure AS or Flex-dependent. All these things are documented in embedded help.

I can't get anything to render on the screen besides the Starling stats

I'm afraid I can't say why your project doesn't work wothout looking at the project. Most likely it contains errors or not configured properly.