Hi Gage,
As far as I can see, you've contacted our TestComplete Support Team, and here is their reply:
The TestedApps and NameMapping project items can be shared among several projects that reside in one project suite. Please make sure that you perform the task exactly as it is described in the "Sharing Project Items and Their Child Elements Among Several Projects" help topic:
If the problem persists, please try to do the same with the two newly created projects and let me know your results.
As for the Stores and Events items, we have a suggestion to implement the ability to share them, and your request has increased its rating. As a workaround, you can clone the existing project, and then include it in your project suite. Please refer to the "Cloning Projects" help topic for details: