Q. Case in point: I have added a project custom field in Company settings > Custom fields. I expect it to be visible in the project details, but I can't find it there. What's going on? 

A: With multiple activity centers, belonging to different spaces, a custom field needs to be part of the respective space or spaces to be available in an activity center and, accordingly, projects and activities that are part of it. A custom field created in Company settings > Custom fields initially does not belong to any space. To make it visible in a particular activity center it needs to be imported there. For details, see Space settings: custom fields & workflows.