A common misconception is that Admins have full access to everythign. This is not true. They are limited to the projects they have been added to, and then are limited to the role they've been given in each project. They only have full access in the Admin zone which is a separate area altogether.
When you select "Create Appointment" in the global tab it checks which projects you have access to, and then determins if your role within each project has the power to create an appointment. It is returning nothing either because that user is a guest in those few system projects or the Calendar application is turned OFF in those few projects or a combo of both.
The other part... Yes, set up a fake project if you only have one project type. That is why we typically roll out Daptiv with 2 project types (Collaborative Workspace is the other) Use collaborative workspaces for non-project info tracking. If you've deleted it, you can create another if you want to differentiate those from regular projects.