1. The screen does initially draw when opening the main window, and screen updates (such as the timer counting down) also draw, but moving the main window blanks it.
2. Bug 28507 was fixed in Wine 1.3.32. This bug still exists in Wine 1.3.32 with my current setup.
3. I've moved from Ubuntu Natty (Gnome) to Ubuntu Precise (Unity).
So, I'm not sure if it should be classed a Wine bug or not. It would appear that the app is not friendly with Unity. I don't know enough about either to hazard a guess.