1. If you happen to experience the same problem, I found that pressing Shift-TAB in the editor makes the pointer visibly darker, and from that point on the tabbing and the Intellisense work correctly. So, this is a bit better, compared to the need to tab through all the controls on the page.
2. Installed Firefox, opened the Hub in Firefox, and everything works well, indeed!
3. Reinstalled Qlik Sense 3.0.1, after declaring Firefox a default browser and rebooting the machine... I was hoping that Qlik Sense Browser could pick up some of the "good stuff" from the default browser, but - no, Qlik Sense Desktop still behaves in the same strange way.
Now I can only wonder - why does it work completely well on one machine, and doesn't work on two other machines.