Is it possible to install CiviCase without all the sample data?
The short answer is no. But it's a good point. I think this is mostly just left over from the original implementation when setting up civicase was difficult, and it was hard to know what it would look like without an example. There didn't used to be a UI - just hand-coded xml files.

You might have to delete some of a case type's definition parts first before deleting the actual type, since it might consider it "in use". But you are going to need at least one case type, so as a workaround for now instead of deleting you can edit the case type.

The relationship that is defined within the case type as the case manager has to exist.

The Open Case activity type does need to exist since it has some semantics associated with it, along with some other activity types, e.g. "Link Cases".

The Urgent status is used for hiliting on the Case Dashboard, but it will function fine if you delete it.