What else apart do I need enabled, apart from "block to region" in Display Suite Extras for the "block regions" vertical tab to appear? I've enabled "block to regions" in Display Suite Extras and I have enabled "Full Content" in "Manage Display".

But the "block regions" vertical tab still does not appear among the other tabs beneath the field table in "Manage Display".

Is there anything I should do to make the "block regions" appear?
You can't use the Default display view mode. You have to enable the Full Content view mode, which will then give you things like Block Regions and custom Page Title options.