Hello,
Going back to your list of requirements, the last point (only having to update something in one place), is EA's basic mode: packages, diagrams and elements are organized in a hierarchy (displayed in the "project browser"), but an element can be shown in any number of diagrams simultaneously and changing any of its properties instantly changes all those diagrams. Put another way, there is no search-and-replace function in EA.
The views you describe could probably be achieved using "auto-color diagram legends". Here's
a short demo. There are three visual properties that can be controlled (fill colour, line colour and line width), and EA will blend the colours if you apply multiple legends to the same diagram. These legends can be configured to react to custom properties, or "tagged values", as well.
The diagram filters Geert also mentioned is a better fit if you want to show/hide things. They're not able to apply custom colours like the diagram legends are.
Relationships are also fundamental in EA, but at its heart it is a UML tool and so by default, all "connectors" are visible. You can switch them off in individual diagrams, but using EA that way is always going to require a bit more manual effort since you're going against the grain.
You can implement custom, non-connector relationships using tagged values. That's not what I would normally recommend a new user to do on day one, but it's certainly doable. Depending on what you then want to do with that information you might need to do some scripting or other adaptations.
But if all you want is to specify that a system belongs to an organizational unit, and is being worked on by a number of projects, and show that information in a diagram without it being a bunch of connectors, then no scripting is required. You can do that with EA Corporate Edition and you can set it up in a matter of minutes.
Well, me or Geert could, anyway.

HTH,
/Uffe