Surprisingly, z-index is not so much used in DayPilotCalendar. It's only used for a few elements (the moving/resizing shadow, hidden events indicators, etc.).
I think the final solution for the printing problems will be a possibility to render a PNG image instead of HTML. But that's not a near future.
You can also try the following hack to improve the IE printing (but I'm afraid it breaks the event borders in Mozilla):
http://forums.daypilot.org/Topic.aspx/74/printing_event_background